calculating parameters for infiltration equations from soil hydraulic functions

25
Transport in Porous Media 24: 315-339, 1996. ~) 1996 Kluwer Academic Publishers. Printed in the Netherlands. 315 Calculating Parameters for Infiltration Equations from Soil Hydraulic Functions P. J. ROSS1, * R. HAVERKAMP 1 and J.-Y. PARLANGE2 l Laboratory for Hydrological and Environmental Sciences (LTHE/IMG), PO Box 53X, 38041, Grenoble, France. e-mail: [email protected] 2Agricultural Engineering, Cornell University, Ithaca, NY, 14853. U.S.A. (Received: 4 July 1995; in final form: 6 February 1996) Abstract. Simple equations for predicting infiltration of water into soil are valuable both for hydro- logical application and for investigating soil hydraulic properties. Their value is greatly enhanced if they involve parameters that can be related to more basic soil hydraulic properties. In this paper we extend infiltration equations developed previously for positive surface heads to negative heads. The equations are then used to calculate infiltration into a sand and a clay for a range of initial and surface conditions. Results show errors of less than three percent compared with accurate numerical solutions. Analytical approximations to parameters in the equations are developed for a Brooks and Corey power law hydraulic conductivity-water content relation combined with either a Brooks and Corey or a van Genuchten water retention function. These are compared with accurate numerical values for a range of hydraulic parameters encompassing the majority of soil types and a range of initial and boundary conditions. The approximations are excellent for a wide range of soil parameters. An important attribute of the infiltration equations is their use of dimensionless parameters that can be calculated from normalised water retention and hydraulic conductivity functions. These normalised functions involve only parameters that it may be possible to estimate from surrogate data such as soil particle size distribution. Application of the equations for predicting infiltration, or their use in inferring hydraulic properties, then involves only simple scaling parameters. Key words: water movement, infiltration, water balance, analytical solutions, hydraulic properties. 1. Introduction Infiltration of water into the soil is an important hydrological process and a con- venient field method of determining soil hydraulic properties. Although an exact description under idealised conditions could be obtained by solving Richards' equa- tion, there are no general exact analytical solutions available and so the process is usually described by a simple relationship between cumulative infiltration and time involving certain physical or empirical soil parameters. Empirical parameters are of limited use and only those with a well-defined physical meaning are generally appropriate. Haverkamp et al. (1992) proposed that two such parameters can be obtained from infiltration experiments, while a third can be derived from the soil particle size distribution. If this is so, then there is real hope of a practical method- ology for estimating the water retention and hydraulic conductivity functions from * Permanent address: CSIRO Division of Soils, Townsville, Queensland, Australia.

Upload: p-j-ross

Post on 06-Jul-2016

219 views

Category:

Documents


3 download

TRANSCRIPT

Page 1: Calculating parameters for infiltration equations from soil hydraulic functions

Transport in Porous Media 24: 315-339, 1996. ~) 1996 Kluwer Academic Publishers. Printed in the Netherlands.

315

Calculating Parameters for Infiltration Equations from Soil Hydraulic Functions

P. J. ROSS1, * R. HAVERKAMP 1 and J.-Y. PARLANGE 2 l Laboratory for Hydrological and Environmental Sciences (LTHE/IMG), PO Box 53X, 38041, Grenoble, France. e-mail: [email protected] 2Agricultural Engineering, Cornell University, Ithaca, NY, 14853. U.S.A.

(Received: 4 July 1995; in final form: 6 February 1996)

Abstract. Simple equations for predicting infiltration of water into soil are valuable both for hydro- logical application and for investigating soil hydraulic properties. Their value is greatly enhanced if they involve parameters that can be related to more basic soil hydraulic properties. In this paper we extend infiltration equations developed previously for positive surface heads to negative heads. The equations are then used to calculate infiltration into a sand and a clay for a range of initial and surface conditions. Results show errors of less than three percent compared with accurate numerical solutions. Analytical approximations to parameters in the equations are developed for a Brooks and Corey power law hydraulic conductivity-water content relation combined with either a Brooks and Corey or a van Genuchten water retention function. These are compared with accurate numerical values for a range of hydraulic parameters encompassing the majority of soil types and a range of initial and boundary conditions. The approximations are excellent for a wide range of soil parameters.

An important attribute of the infiltration equations is their use of dimensionless parameters that can be calculated from normalised water retention and hydraulic conductivity functions. These normalised functions involve only parameters that it may be possible to estimate from surrogate data such as soil particle size distribution. Application of the equations for predicting infiltration, or their use in inferring hydraulic properties, then involves only simple scaling parameters.

Key words: water movement, infiltration, water balance, analytical solutions, hydraulic properties.

1. Introduction

Infiltration of water into the soil is an important hydrological process and a con- venient field method of determining soil hydraulic properties. Although an exact description under idealised conditions could be obtained by solving Richards' equa- tion, there are no general exact analytical solutions available and so the process is usually described by a simple relationship between cumulative infiltration and time involving certain physical or empirical soil parameters. Empirical parameters are of limited use and only those with a well-defined physical meaning are generally appropriate. Haverkamp et al. (1992) proposed that two such parameters can be obtained from infiltration experiments, while a third can be derived from the soil particle size distribution. If this is so, then there is real hope of a practical method- ology for estimating the water retention and hydraulic conductivity functions from

* Permanent address: CSIRO Division of Soils, Townsville, Queensland, Australia.

Page 2: Calculating parameters for infiltration equations from soil hydraulic functions

316 P.J. ROSS ET AL.

infiltration measurements. The work reported here seeks analytical expressions for infiltration which involve parameters related (analytically) to those in common forms of the soil water retention and unsaturated hydraulic conductivity functions. We will show that some hydraulic property parameters enter the analysis very sim- ply as scaling parameters; others involved in a more complex way could reasonably be estimated from surrogate data such as soil particle size distribution.

This paper considers infiltration into a uniform soil from a surface source at a constant head. Philip (1969) reviewed early work in this area. Parlange e t al. (1982) presented an equation expressing cumulative infiltration implicitly in terms of three physically-based parameters: sorptivity, conductivity and a parameter related to the form of the hydraulic property functions. More recently, Parlange e t al. (1985) and Haverkamp e t al. (1990; 1995) considered sources at positive heads. This paper extends the previous work, giving a concise description applicable to both positive and negative heads. It presents analytical expressions for the required parameters in terms of common soil water retention and unsaturated hydraulic conductivity functions and demonstrates that the expressions are accurate over a wide range of initial and surface boundary conditions and soil types. In addition, it demonstrates that the new infiltration equation using these expressions is accurate for a sand and a clay. Since three-dimensional infiltration from a disk has recently been described using one-dimensional infiltration equations as a basis (Haverkamp e t al. , 1994), we expect these expressions to be useful for three-dimensional infiltration also.

The analysis is based on the one-dimensional Richards equation which, though amply verified for uniform soil material, is less than perfect for describing infiltra- tion in heterogeneous field soils. Currently, however, there appears to be no better alternative.

2. T h e o r y

We assume that infiltration is described by Richards' equation

oo rO ' )] Ot' = Oz I \ Oz ' - I ,

(1)

where 0 is volumetric water content, t t is time, z ~ is depth, K is hydraulic con- ductivity and h' is matric head. We will work throughout this paper in terms of dimensionless variables defined by

0 - Or K # ' z' O = ~ t = z = - -

Os -- Or' L(Os - Or) ' L '

k = K h' Ks ' h = -~-,

(2)

Page 3: Calculating parameters for infiltration equations from soil hydraulic functions

INFILTRATION EQUATIONS FROM SOIL HYDRAULIC FUNCTIONS 317

where Or and Os are residual and saturated water contents, K s is saturated con- ductivity and L is a positive reference length. For the van Genuchten (1980) water

where h i is the head scaling parameter, giving retention curve we choose L = - h g ,

O = 1 + = [1 + (-h)'~] m, (3)

where m and n are soil constants. For the Brooks and Corey (1964) relation, we take L = -h~, where h~ is the bubbling pressure, giving for h ~ < h~

O = = ( - h ) - a , (4)

where A is a soil constant. Applying the above scaling to soil diffusivity D ~, sorptiv- ity S '~ and cumulative infiltration I ~ gives corresponding dimensionless diffusivity, sorptivity and infiltration

S' I ' D -- (Os - Or)D' S =- I - (5) L K s ' [LKs(O, - Or)] I / z ' L(Os - Or)"

In terms of the scaled variables, Richards' equation becomes

0 0 0

We consider infiltration into a soil at uniform initial matric head h0 from a surface boundary source at constant head hi. The subscript 0 will refer to initial conditions and subscript I to surface boundary conditions. The soil becomes saturated at a matric head hs <~ O. For this problem h decreases with depth z from its value hl at the surface to h0 at the wetting front. Changing to independent variables h and t, Richards' equation becomes

d-h ~" = d O Oz k / ~ - ~ . ~ (7)

and integrating from ho to h gives

Oz qo - q = k / - ~ + ko - k, (8)

where q is the dimensionless flux at h at time t. Solving for Oz/Oh and integrating between hi and h wef ind

fh hI k dh z = k o - k + q - qo"

(9)

Page 4: Calculating parameters for infiltration equations from soil hydraulic functions

318 P.J. ROSS ET AL.

Integrating z by parts from Oo to 01 we have

L hl (0 - - O0)k dh fO 9' (0 -- Oo)D dO, I - kot = o ko - k + q - qo = o ~'-"--k---+ q Z--qo (lO)

where D is a dimensionless diffusivity defined by

dh D = k - ~ + (hl - h~)5(O - 1 ) (11)

with the Dirac delta function 5(0 - 1) accounting for saturated conditions (hi >/ h2.

Parlange et al. (1982) introduced a three parameter infiltration equation wherein one of the parameters fl(0 <~ fl ~< 1) interpolated between a Green and Ampt type of equation, where dk/dO increases much less rapidly with O than the diffusivity (fl ..~ 0), and an equation derived for similar behaviours of dk/dO and the diffusivity (fi ~ 1). Haverkamp etal. (1990) generalised slightly the relation assumed between conductivity and diffusivity to give

k ( o ) - k0 k~ - ko

2fl O 1 0 - Oo - dO] -f(O) [1- ~ fo ~-~ D~(O) , (12)

where Du is diffusivity and f(O) = F(O, 0), where F(O, t) = (q - qo) / (ql - qo) was named the 'flux-concentration relation' by Philip (1973). The subscript u indicates the unsaturated region. Su is the sorptivity given by

fo ~10-O~ d 0 Sua=2 f(O) " 0 (13)

The work of Haverkamp et al. (1990) was concerned with positive head infiltration (hi /> 0). These authors took Du as the usual diffusivity in the unsaturated region and allowed for the effects of hs < 0 by adding an appropriate delta function. An infiltration equation was then derived where hs was treated as an unknown infiltration parameter. However, since this equation was derived by integration using the assumption about fl given above, the estimate of hs in fact reflected the large values of slope dh/dO of the water retention curve which are often found below but near to saturation and which make direct determination of hs difficult. Thus, for example, the van Genuchten (1980) retention curve, with zero hs, resulted in a positive estimated value when combined with a Brooks and Corey (1964) power law hydraulic conductivity (Haverkamp et al., 1995). Here we make use of this behaviour to extend the methods to unsaturated surface conditions hi < hs by introducing a similar head parameter ha. We do this through the relation

D ( O ) = D u ( O ) + k l h a : ( O - 01) , (14)

Page 5: Calculating parameters for infiltration equations from soil hydraulic functions

INFILTRATION EQUATIONS FROM SOIL HYDRAULIC FUNCTIONS 319

where the second term is a delta function which approximates the contribution of the large slopes to the diffusivity. As hi decreases, the slopes become smaller and this contribution will approach zero. Both Du and ha are initially unknown; while hd can be estimated, Du will only be used in integrals and is never otherwise required. Note that, where hi > hs, hd includes (hi - hs).

Using the foregoing relations, it is possible to integrate useful expressions involving the variable

k ( O ) - k o v - if(O), (15)

kl -- kO

since, if u(v) is any function, we have using Equation (12)

fo~' u ( v ) ~ D ( O ) dO

/;1(.) = - o u 1- - -~uy d y + k l h d ( O l - O o ) u ( 1 ) , (16)

where

y = f:~ (~ - Oo - -f(~ Du(e) dO. (17)

In particular, since q - qo = (ql - qo)F(O, t) and F(O, t) ~ F(O, 0) = f(O) (Philip, 1973), taking u(v) = 1/[-(ki ko)v + qs - ko] leads from Equation (10) to the infiltration equation (Haverkamp et al., 1990)

I, __ _ _

t * - -

o-

q~ - 1 1 -o - [ ,/3

+ - - - ~ In l + - - ql - 1

1--0"

/3(1 -/3) In [1 + q ~ _ 1 o- 1 - o - / 3 [ 1 ]

+ q ~ l ~ l n 1 + ~ ,

(18)

where q~ -- dI*/dt* and

I* -~ 2(kl -- k ~ kOt)

O- -~-

S 2

2(kl - k o ) 2 t $2

2klhd(O1 - 0o) $2

(19)

S 2 = S2u + 2klhd(O1 - 00).

Page 6: Calculating parameters for infiltration equations from soil hydraulic functions

320 l,. J. ROSS ET AL.

The equations relate I* and t* implicitly with q~ as a parameter. N6te the relation between ~r and ha which shows that a represents the effect of the delta function components of the diffusivity, as discussed previously for hd. Thus we expect a to be small or zero with an unsaturated upper boundary condition hi and to increase as hi approaches zero and becomes positive. The above equations cannot be used when/3 = 1 (similar behaviours of dk/dO and the diffusivity); instead the limiting form

[ ,] q ~ , ~ + ( 1 - a ) l n l+q--~--~_ 1 ,

1 t * = ( 1 - 2 a ) ln 1+

a 1 - a + - - q ~ - 1 q~

(20)

of the infiltration equations must be used. To estimate S,/3 and a we derive relationships using functions that can be

integrated analytically. Taking u(v) = 1 in Equation (16) shows that

fo % 0 - 0OD(o) dO. ,5'2= 2 f(O) 0

(21)

Thus S is the sorptivity which can be evaluated analytically given a suitable D(O) and a suitable approximation for f(O). The capillary rise equation

/cap = fo~' k (0 ) -- k00-----~~ D ( 0 ) d O (22)

obtained from Equation (10) by setting qs - k0 = 0 leads to taking u(v) = 1Iv in Equation (16) giving

k l - k o S 2 [ ( 1 - a ) ln (1- /3) ]

- OoZ P - 2 ( o l o0) /3 01 _ o r - �9 , (23)

where the maximum cumulative capillary rise at infinite time,/cap, can be evaluated analytically from Equation (22) given appropriate soil hydraulic functions k(O) and D(O). Note that/cap becomes infinite when 0o = 0, and then fl = 1. Similarly, the integral

)tK = 2 /,a 9~ k(O) - ko O - Oo D(O)

kl - kO.vo -ko o l - Oo f ( o ) 2 d ~

2 f g , O__ ~.O.qo D(O),tA ~1 -- k o Joo v 01 - Oo f(O) - v ,

(24)

Page 7: Calculating parameters for infiltration equations from soil hydraulic functions

INFILTRATION EQUATIONS FROM SOIL HYDRAULIC FUNCTIONS 321

can be approximated analytically, and also leads to the relation

S 2

(ks - kO)AK = [2 - /3(1 - a)]2(Oi _ Oo)' (25)

when we take u(v) = v in Equation (16). Note that, from Equations (24) and (21), (kl - k0)AK is the same as $2 / (01 - O0) except that the integrand is weighted by v. Note also that, when/3 = 1 (e.g. when O0 = 0), (kl - ko)AK = (1 + cr)S2/2(@l - O0) = S~/2(O1 - O0) + 2klhd (see Equation (19)). If in addition ko is negligible, An- = $2 /2k l (01 - 6)0) + 2ha, i.e. AK is simply related to ha.

The parameters/3 and a can therefore be determined from the soil hydraulic property functions by evaluating leap and AK and solving Equations (23) and (25). If the solution for cr should be negative, we set cr --- 0 (since ha /> 0) and solve Equation (23) for/3. The method fails when/3 approaches zero, i.e., when f(O) approaches [k(O) - ko]/(kl - ko) (see Equation (12)), since it is clear from Equations (23) and (25) that cr cannot be found when/3 = 0. In fact, errors in estimates of leap and AK can cause problems before/3 reaches zero, since solution then depends on differences between almost equal quantifies. Fortunately, this is not likely to be a problem for most soils, because k usually decreases much faster with (9 than does f(O). When fl = 0, substituting f(O)(k~ - k0) for k(O) - ko in Equation (10) allows immediate integration to obtain as expected the Green and Ampt equation

s2 [1+ 2(kl _ ko) ii( ) _ kotj ] k i t = I ( t ) - 2 ( k l - k0)In $2 . (26)

To relate the above theory to soil hydraulic parameter estimation, we note that the dimensional quantities S ~, F and t ~ are given in terms of the dimensionless S, I and t by Equations (2) and (5) while/3 and o- are dimensionless. These relations are important for estimating hydraulic properties by inverse procedures because they show clearly that some parameters simply scale S 2,1 and t, and do not occur at all in/3 and a, whatever particular forms of the hydraulic functions may be chosen. Only parameters determining the relations between the dimensionless quantities O, h and k remain in the analysis; these parameters are interrelated and can often be estimated from surrogate data such as particle size (Haverkamp et al., 1992; Smettem et aL, 1994).

2.1. SPECIFIC HYDRAULIC FUNCTIONS

Using the approximation (Parlange, 1975; Haverkamp et al., 1995)

2(o-o0) f(o) = 0 +-01 - 0o (27)

Page 8: Calculating parameters for infiltration equations from soil hydraulic functions

3 2 2 P.J. ROSS ETAL.

and defining

f o (')1 J(o 0 = 19"D(19) d19, (28) 0

where it is understood that J (o 0 also depends upon 190 and 191, we adopt a power law for the conductivity function (Brooks and Corey, 1964)

k(19) = 19,1, (29)

which can provide a range of behaviour as ~ varies. It is usually accepted that r / > 3 (Fuentes et al., 1992). Using the approximation

1 - - r r/ - - -- 1 q- r + . . . q- r j - 1 + (rj -- j ) r j (30) 1 - - r

with j the largest integer less than r/to express the factor

k ( e ) - ko 1 -- ( 190 /0 ) r / ~ - - 1 19 - 19o - i - - ~ ~ ' ' (31)

in the expression for ),h" in powers of 190/19, integration to obtain S a,/cap and AK is straightforward, giving

S 2 -- J(1) + c J(0),

i~-(3o

Icap = ~ 19~n{J[1- (i + 1) r / ] - 190J[-(i + 1)~7]}, (32) i=0

2 j ( k l - k o ) A K = -7- ~ (c2ai + 2ca i+l + a i + 2 ) J ( ~ - 1 - i),

a e i = - 2

where

c = (91 - 200, d = 2(191 - 19o), e = 2(kl - ko),

a - 2 -: a - 1 ~- a j + l ~ a j+2 -- O~ a 0 ~- l ,

ai = 1 9 1 , O < i < j , (33)

a j = ( ~ - i)og. The approximation of Equation (30) has an error less than 0.7 percent for r/ > 3 and is exact for integer values of rl. For 19o = 0 the value AKO of AK is given exactly by

2 klAgo = ~ee [g ( r /+ 1) + 2cJ(r/) + c 2 g ( ~ - 1)]. (34)

Page 9: Calculating parameters for infiltration equations from soil hydraulic functions

INFILTRATION EQUATIONS FROM SOIL HYDRAULIC FUNCTIONS 323

Furthermore, because conductivity and diffusivity fall sharply with water content, (kl - k0)),g is almost independent of initial condition 190 below O0 values of 0.5 to 0.7, depending on the soil, and kl),K0 itself provides an excellent approximation.

We consider two water retention functions: that of Brooks and Corey (1964) and that of van Genuchten (1980).

(1) Water Retention Function of Brooks and Corey (1964)

We have

O = ( - h ) -h, h ~ < - l , (35)

where saturation occurs at h ----- -1. Then

D(O) = 1AO~_I/A_I + (hi + 1)6(0 - 1),

J(ce) - ~/~A1/A (O?+rl-1/A -o~+~-I/A) + (hl + I)6(OI - 1 ) ty+

(36)

and

~K0 --- ~ ~ + ~ + + 2 (< + 1)a(oi - 1),

b = 2 r / - 1/A.

(37)

(2) Water Retention Function of van Genuchten (1980)

We have

O = [l + (-h)n] m, h~<O,

where saturation occurs at zero head. Then

D(O) = 10~l_l/mn_l(1 _ o1/m)l/n_l d- hlt~(O - 1), mn

n n

where/3[] is the usual incomplete beta function, and

~KO -- 2nOlk 2 1 [Boll/m ( ( b + 1)m, 1 ) + 201Boll~ m (bin, ~) q-

(38)

(39)

Page 10: Calculating parameters for infiltration equations from soil hydraulic functions

324 P.J. ROSS El" AL.

1 b = 2r j - - - .

r a n

(40)

3. Materials and Methods

3.1. SAND AND CLAY

The soils used for illustration were Grenoble sand (Touma et al., 1984) and Yolo light clay (Moore, 1939) with parameters (Table I) as given by Fuentes et aL (1992). Only the van Genuchten water retention function was used. These soils show contrasting behaviour for infiltration (Haverkamp et al., 1977). A set of 'accurate' and a set of 'approximate' infiltration parameters were calculated as follows. Values of S were calculated accurately by numerical integration of the Bruce and Klute (1956) integro-differential equation using an iterative process and the NDSolve command of Mathematica (Wolfram Research, Inc., 1992). This procedure also gave f(O). Icao was calculated accurately using Equation (32). Accurate values of AK were obtained from numerical integration (Equation (24)) using the accurate f(@). Accurate values of/3 and a could then be calculated from Equations (23) and (25). Approximate values of S and ),K (and an accurate/cap) were calculated from Equation (32) using the approximate f(O) of Equation (27), giving approximate values of fl and a. Infiltration was calculated from Equation (18) and also by numerical integration of Richards' Equation (6) using finite differences (Ross and Bristow, 1990) with discretisation and mass balance criteria chosen to keep errors negligible. The final time was chosen to be 5tgrav, where tgrav = $2/ (k l - k0) 2 gives an estimate of the time at which the effect of gravity becomes comparable with that of capillarity (Philip, 1969). Note that t* = 2t/tgrav, where t* is the dimensionless infiltration time defined in Equation (19). Values of the initial condition 190 were taken to cover the range 0 to 0.5 while the surface boundary conditions were Oj = 0.9 and 191 = 1 with h~ = 0 and hi = 1.

3.2. PARAMETERS REPRESENTING OTHER SOILS

Values of the van Genuchten parameter m were taken to cover the range expected for the majority of soils while n was given by the relation n = 2/(1 - m) recommended by Fuentes et al. (1992). Corresponding values of r/were taken slightly above the minimum where the diffusivity became infinite at r/ = � 89 + 1) and also at approximately twice the value ~/ = 1 / m + 2 used by Fuentes et al. (1992). In addition, a lower limit of 3 was imposed (Fuentes et al., 1992). For the Brooks- Corey water retention function, values of A were taken equal to ran since the van Genuchten function tends towards the Brooks-Corey with A = ran as the water

Page 11: Calculating parameters for infiltration equations from soil hydraulic functions

INFILTRATION EQUATIONS FROM SOIL HYDRAULIC FUNCTIONS 325

Table I. Soil hydraulic parameters

Grenoble sand Yolo light clay

0~ 0,312 0.495 0T 0.0 0.0 Ks 15.37cmh -1 0.0443 cmh -I h i -16.39cm -19.31 eva n 2.793 2.221 m 0.2838 (1 - 2/n) 0.0995 (1 - 2In) 7/ 6,728 9.143

content decreases. S,/cap, ~K, fl and cr were calculated as for the sand and clay for the same initial and boundary conditions.

4. Results and Discussion

4.1. SAND AND CLAY

4.1.1. Comparison of Accurate and Approximate S, )~K, fl and a

Figures 1 to 4 show, for the sand and the clay, the accurate infiltration parameters S, ),K, fl and cr and their differences from the approximate parameters. Results are given as a function of initial condition O0 and the surface boundary condition. All parameters except fl change sharply as the surface boundary condition approaches saturation. The analytical approximation for S is very good, with errors of approx- imately one percent, while approximations for AK, [3 and o are sufficiently good to introduce little additional error into the calculation of infiltration. S 2 is very nearly linearly related to O0 [Philip, 1957], giving a slightly curvilinear relation between S and Oo. Note that , ~ ~ XK0 over a wide range of initial water contents, allowing Eq. [40] to be used. Further investigation showed that, as might be expected from the definition of)~r, the approximation (kl - k0) A/~ = kl AK0 was usually superior when ko was appreciable. As expected, cr falls sharply towards zero below satura- tion;/3 remains close to 1 for all conditions, although it is decreasing substantially at the initial water content Go = 0.5.

4.1.2. Comparison of Predicted Infiltration with Numerical Solution

Values of the accurate parameters at selected initial and boundary conditions used for infiltration calculations are given in Table II. Figure 5(a, b) shows for the sand and the clay an example of infiltration calculated from accurate and approx- imate parameters compared with infiltration obtained from numerical solution of Richards' equation. There is almost no difference in the results. Water content profiles from the numerical solution are shown in Figure 5(c, d) and accurate ver-

Page 12: Calculating parameters for infiltration equations from soil hydraulic functions

326 P.J. ROSS ET AL.

(a)

1 1

o. ~ o ' 2 G, 0 3 6),

" 0.5 ~'- (b )

~ 0 ~ " 0 . 5 u . ~

Error

0"010~

_0.011 I " 1 -0.02w_..

~.- o.3 " - ~ ~Q 0.4 0.9 0.5

E r r o r ~ 0.0

-O_oO

Figure 1. Dimensionless infiltration parameters for Grenoble sand and absolute errors in analytical approximations; (a) sorptivity S; (b) AK. O0 is the initial normalised water content while the surface boundary condition is O1 below saturation and hi above saturation.

sus approximate f(O) in Figure 5(e, f). The approximation for f(O) is good in this case; for wetter surface boundary conditions the approximation at the wet end improves at the expense of that in the drier range where it has little effect on the infiltration parameters. Figures 6 and 7 show, for the sand and the clay, percent errors (based on the accurate numerical solution of Richards' equation) in pre- dictions of cumulative infiltration given by the infiltration equations with accurate and approximate parameters. Errors for both accurate and approximate parameters are less than two percent, except for infiltration into the dry clay with a surface boundary condition of O1 -- 0.9, when cr = 0 and fl = 1 (Table II). The maximum error is then approximately three percent. Errors at early times are greater with the approximate parameters than with the accurate ones, reflecting mainly small errors in sorptivity. The prediction of infiltration for these two soils over the given range of times is therefore accurate enough for practical purposes such as helping to estimate hydraulic properties.

Page 13: Calculating parameters for infiltration equations from soil hydraulic functions

INFILTRATION EQUATIONS FROM SOIL HYDRAULIC FUNCTIONS 327

(a)

E r r o r ~

A o.o2 0.9 ! 0 . 0 1 0 ~

�9 0 . 2 0 ~ 01 O, 0.." 6) 0 0.4 0.5 0.9 " "

(b)

(y E r r o r ~

0 ~

001 oh: : 0 o 0.4 0.9 ~0. 0.4 0.~ 0.9

0.5

Figure 2. Dimensionless infiltration parameters for Grenoble sand and absolute errors in analytical approximations; (a)/3; (b) a. O0 is the initial normalised water content while the surface boundary condition is Ol below saturation and hi above saturation.

4.1.3. A UsefuI Approximation

Although both fl and a can be calculated from the equations given, it is of interest to see whether reasonable accuracy in predicting infiltration can be achieved with one less parameter, as in the equation of Parlange et al. [1982]. Since fl = 1 for initial water contents O0 = 0 and remains close to 1 for other conditions, it seemed that setting fl = 1 might provide a reasonable approximation, as Haverkamp et al. (1990) found. In addition, (kl - k0)AK is approximated very well by klAKo for the initial water contents investigated here, and AK0 is readily calculated from Equation (40). Therefore, cr (limited to zero or positive values) was obtained from Equation (25) with/3 = 1 and AK = klAKo/(kt - ko), thus simplifying the calculations considerably, and used with the infiltration equations (20). Results with this combination were, of course, identical for O0 = 0, but even for Oo = 0.5 only slight increases in the errors occurred, with the sand at O1 = 0.9 being most affected, the maximum error there rising to 2.9 percent for the accurate parameters and 3.2 percent for the approximate ones. For these soils and conditions, then, the

Page 14: Calculating parameters for infiltration equations from soil hydraulic functions

328 V.J. ROSS ET AL.

3

12

4

r

vQ

(a)

0.5 t#.y

(b)

~0 =" 0.5 u.~

E r r o r ~

OoV- I / ~ (9 ~ 0.3 0 .4~05 0.9 ~,

E r r o r ~

o h �9

/ / ~

0,, ~'~ 0.4 0 . ~ 0 9

Figure 3. Dimensionless infiltration parameters for Yolo light clay and absolute errors in analytical approximations; (a) sorptivity S; (b) AK. O0 is the initial normalised water content while the surface boundary condition is O1 below saturation and hi above saturation.

approximation is acceptable. A further advantage is that an accurate explicit form of Equations (20) has been developed (Barry et al., 1995).

4.2. PARAMETERS REPRESENTING OTHER SOILS

Tables III and IV give values of the accurate infiltration parameters, together with the errors in the approximate values, for the van Genuchten water retention function with m and r/values chosen to cover a wide range of soil types.

4.2.1. Values of S and IK

As expected, S 2 is almost linearly related to initial water content 00. Errors in the approximate value of S are no larger than 1.1 percent except for the smaller values of ~/with an unsaturated boundary water content of O1 = 0.9 when the error increases to a worst case value of 5.2 percent. These smaller values of r/are extreme, being close to the limit (1/m + 1)/2 at which the diffusivity becomes infinite. Similar remarks apply to errors in ),K, although they are larger than

Page 15: Calculating parameters for infiltration equations from soil hydraulic functions

INFILTRATION EQUATIONS FROM SOIL HYDRAULIC FUNCTIONS 329

(a)

Error~~~__~ 0 1

O.9 ' oh; o /1 0 .I 0.2 ^ ~ ~ l

O . 0.4 0.9 0.5

(b)

0.5 u.~, --O v~,

(Y

O.

0

E r r o r ~ 0.04

0.02 1

0.3 0 ~ 0 . 9 - ' 0,5

Figure 4. Dimensionless infiltration parameters for Yolo light clay and absolute errors in analytical approximations; (a) fl; (b) ~r. O0 is the initial normalised water content while the surface boundary condition is O1 below saturation and hi above saturation.

those in S; however, since these errors do not lead to corresponding errors in/3 and or, they have little effect on calculated infiltration. Note that the approximation AK = kt AKo/(kl -ko), with AK0 calculated analytically, is an acceptable substitute for the approximate AK except when O1 = 0.9 and r] = 3; in this case it is preferable to take AK = AKO.

4.2.2. Values of fl and

The value o f /3 is 1 when O0 = 0, as mentioned previously. In fact, /3 only differs substantially from 1 for r~ i> 0.25 and either a small value of ~7 or wet initial conditions or both. It becomes small, with a large error in the analytical approximation, only for m = 0.5, r/ = 3 and O0 = 0.5. For many soils and conditions a value of 1 could be assumed without serious error, as suggested by Haverkamp et aL (1990) for ponded infiltration. As expected, cr is small for O1 = 0.9 and increases with the head hi at the upper boundary. Errors in the analytical approximation are small, except for rn - 0.5 with r / = 3.

Page 16: Calculating parameters for infiltration equations from soil hydraulic functions

330

Table II. Infiltration parameters

E J. ROSS ET AL.

Grenoble sand

Oo 0 0 0 0.5 0.5 0.5 Ol 0.9 1 1 0.9 1 1 hi -0.75 0 1 --0.75 0 1 S 0.57 1.26 1.90 0.36 0.88 1.34 /cap c~ c~ c~ 0.39 1.36 2.36 ),K 0.39 1.18 3.20 0.39 1.20 3.26 ha 0.03 0.38 1.39 0.03 0.37 1,38 fl 1 1 1 0.91 0.88 0.87 tr 0.07 0.48 0.77 0.08 0.47 0.77

Yolo light clay

O0 0 0 0 0.5 0.5 0.5 O1 0.9 1 1 0.9 1 1 hi -1.33 0 1 -1.33 0 1 S 1.03 1.75 2.26 0.62 1.19 1.57 /cap e~ c~ c~ 2.76 7.02 8.02 AK 1.17 1.67 3.70 1.13 1.67 3.73 ha 0 0.14 1.14 0 0.25 1.26 fl 1 1 1 0.99 1.00 1.00 a 0 0.09 0.45 0 0.18 0.51

4.2.3. Brooks and Corey Water Retention Function

Tables V and VI present results for the Brooks and Corey water retention function. These are similar to those for the van Genuchten function. Errors in S are slightly higher, with a maximum of 1.4 percent for S, except for smaller values of r /wi th an unsaturated boundary, where the error rises to 7.3 percent. Errors in )~x( are also slightly higher. Values of fl are similar, although slightly higher with smaller errors. The parameter a expresses the effect of saturation and near saturation, as given by Equation (19). As might be expected from the water retention curves, with a tension-saturated zone for the Brooks and Corey function, cr is considerably larger than for the van Genuchten function for applied heads hi /> 0 but almost zero for unsaturated conditions O1 = 0.9. From Equation (19), values of a close to 1 occur when the unsaturated component of sorptivity squared, S 2, is much smaller than the near-saturated and saturated component, 2kt ha (O1 - O0). Errors in analytical approximations of a are smaller than for the van Genuchten function.

Page 17: Calculating parameters for infiltration equations from soil hydraulic functions

Tab

le II

I. V

alue

s of

S a

rid

AK

with

per

cent

err

ors

of a

naly

tica

l app

roxi

mat

ions

for

van

Gen

ucht

en w

ater

ret

enti

on fu

ncti

on

S E

rror

per

cent

A

K

Err

or p

erce

nt

m

r/

Oo

O1

=0

.9

ht=

0

hl=

l O

1=

0.9

h

i=0

h

l=l

O1

=0

.9

hi=

0

h1

=1

O

1=

0.9

h

i=0

h

1=

1

0.05

13

0.

00

1,72

2.

48

2.88

1.

0 0.

0 -0

.8

3.11

2.

21

4.24

3.

9 1.

7 0.

3 0.

25

1.41

2.

10

2.46

1.

9 0.

3 -0

.8

3.07

2.

20

4.24

5.

5 2.

3 0.

3 0.

50

1.00

1.

62

1.93

3.

8 1.

0 -0

.6

3.00

2.

19

4.24

8.

9 3.

2 0.

3

44

0.00

0.

12

1.21

1.

86

0.0

-0.1

-0

.1

0.69

0.

95

2.96

0.

5 0.

2 -0

.1

0.25

0.

10

1.05

1.

61

0.0

-0.1

-0

.1

0.69

0.

95

2.96

0.

7 0.

1 -0

.1

0.50

0.

08

0.85

1.

32

0.0

-0.2

-0

.2

0.69

0.

95

2.96

1.

1 0.

2 -0

.2

0.10

7

0.00

1.

47

2.10

2.

56

1.7

0.0

-1.0

1.

61

2.02

4.

07

5.9

2.1

-0.1

0.

25

1.18

1.

76

2.17

3.

0 0.

5 -1

.0

1.59

2.

02

4.08

8.

2 2.

7 -0

.2

0.50

0.

81

1.34

1.

70

4.7

1.1

-0.9

1.

53

2.02

4.

14

13.0

3.

5 -0

.7

24

0.00

0.

25

1.18

1.

84

0.0

-0.1

-0

.2

0.40

0.

94

2.95

0.

8 0.

1 -0

.2

0.25

0.

21

1.02

1.

59

0.0

-0.2

-0

.2

0.40

0.

94

2.95

1.

1 0.

1 -0

.2

0.50

0.

16

0.83

1.

30

0.1

-0.3

-0

.3

0.39

0.

94

2.95

1.

9 0.

1 -0

.4

0.25

3

0.00

1.

18

1.73

2.

26

3.0

0.1

-1.2

0.

99

1.87

3.

95

9.5

2.0

-1.2

0.

25

0.91

1.

43

1.91

4.

3 0.

4 -1

.2

0.98

1.

90

4.05

13

.0

2.1

-1.9

0.

50

0.61

1.

08

1.49

5.

2 0.

5 -1

.3

0.95

2.

02

4.47

17

.2

2.1

-2.8

12

0.00

0.

36

1.13

1.

81

0.0

--0.

2 -0

.2

0.26

0.

95

2.96

1.

3 0.

2 -0

.3

0.25

0.

30

0.98

1.

57

0.1

-0.3

-0

.3

0.26

0.

95

2.96

1.

9 0.

1 -0

.4

0.50

0.

23

0.79

1.

28

0.2

-0.5

-0

.5

0.26

0.

95

2.97

3.

1 -0

.1

-0.7

0.5

3 0.

00

0.66

1.

33

1.95

0.

6 -0

.3

-0.5

0.

40

1.46

3.

49

4.3

0.4

-0.7

0.

25

0.54

1.

14

1.68

1.

3 -0

.4

-0,7

0.

40

1.49

3.

57

6.1

-0.2

-1

.3

0.50

0.

38

0.91

1.

36

2.7

-0.4

-0

.8

0.42

1.

64

4.00

10

.2

-0.7

-2

.1

8 0.

00

0.35

1.

14

1.82

0.

0 0.

5 0.

1 0.

18

1.08

3.

09

1.6

1.7

0.3

0.25

0.

29

0.99

1.

57

0.1

0.4

0.0

0.18

1.

08

3.10

2.

2 1.

5 0.

2 0.

50

0.22

0.

80

1.28

0.

4 0.

1 -0

.2

0.18

1.

09

3.12

3.

7 1.

0 -0

.2

t..n

:Z

r-

z q

Page 18: Calculating parameters for infiltration equations from soil hydraulic functions

Tab

le I

V. V

alue

s of

fl a

nd c

r w

ith

erro

rs o

f an

alyt

ical

app

roxi

mat

ions

for

van

Gen

ucht

en w

ater

ret

enti

on f

unct

ion

to

Err

or

cr

Err

or

m

T]

O0

O1

=

0.9

hl

=

0 ht

=

1 O

l =

0.

9 hi

=

0 hx

=

1 O

1 =

0.9

hi

=

0 hi

=

1 O

1 0

.9

hi =

0

hi

=

1

0.05

13

0.

00

1.00

1.

00

1.00

0.

00

0.00

0.

00

0.00

0.

00

0.02

0.

00

0.00

0.

02

0.25

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.00

0.

05

0.00

0.

00

0.02

0.50

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.00

0.

14

0.00

0.

00

0.02

44

0.00

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.31

0.

71

0.00

0.

00

0.00

0.25

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.31

0.

71

0.00

0.

00

0.00

0.50

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.31

0.

71

0.00

0.

01

0.00

0.10

7

0.00

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.00

0.

24

0.00

0.

00

0.02

0.25

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.00

0.

29

0.00

0.

00

0.02

0.50

0.

99

0.99

0.

99

-0.0

1

0.00

0.

00

0.00

0.

11

0.42

0.

00

0.01

0.

02

24

0.00

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.36

0.

74

0.00

0.

01

0.00

0.25

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.36

0.

74

0.00

0.

01

0.00

0.50

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.37

0.

74

0.00

0.

01

0.00

0.25

3

0.00

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.25

0.

54

0.00

0.

02

0.02

0.25

0.

93

0.92

0.

90

-0.0

2

0.00

0.

02

0.02

0.

32

0.60

0.

03

0.02

0.

02

0.50

0.

78

0.73

0.

68

-0.0

6

0.00

0.

06

0.05

0.

35

0.64

0.

02

0.02

0.

02

12

0.00

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

05

0.49

0.

80

0.01

0.

01

0.00

0.25

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

06

0.49

0.

80

0.02

0.

01

0.00

0.50

0.

98

0.97

0.

97

0.00

0.

00

0.01

0.

08

0.50

0.

81

0.03

0.

02

0.02

0.5

3 0.

00

1.00

1.

00

1.00

0.

00

0.00

0.

00

0.23

0.

65

0.84

0.

04

0.02

0.

01

0.25

0.

84

0.73

0.

71

0.00

0.

08

0.10

0.

16

0.57

0.

80

0.05

0.

06

0.03

0.50

0.

70

0.47

0.

38

-0.0

3

0.18

0.

26

0.11

0.

45

0.72

0.

05

0.16

0.

10

8 0.

00

1.00

1.

00

1.00

0.

00

0.00

0.

00

0.16

0.

66

0.87

0.

02

0.01

0.

00

0.25

0.

98

0.95

0.

94

0.00

0.

01

0.01

0.

16

0.65

0.

86

0.02

0.

02

0.00

0.50

0.

90

0.82

0.

81

0.01

0.

05

0.06

0.

13

0.62

0.

85

0.04

0.

04

0.01

.r-

Page 19: Calculating parameters for infiltration equations from soil hydraulic functions

Tab

le V

. Val

ues

of S

and

AK

with

per

cent

err

ors

of a

naly

tica

l app

roxi

mat

ions

for

Bro

oks

and

Cor

ey w

ater

ret

enti

on fu

ncti

on

S E

rror

per

cent

),

K

Err

or p

erce

nt

A

7/

O0

O1

=0

.9

hi=

0

hl=

l O

1=

0.9

h

i=0

h

l=l

O]=

0.9

h

i=0

h

l=l

Oa

=0

.9

hi=

0

hl=

l

:2

0.10

5 13

0.

00

1.71

2.

63

3.00

1.

1 -0

.5

-0.9

3.

02

3.14

5.

16

4.0

0.4

-0.1

0.

25

1.41

2.

24

2.57

1.

9 -0

.4

-1.1

2.

98

3.14

5.

16

5.7

0.6

-0.1

0.

50

0.99

1.

74

2.02

3.

9 -0

.1

-1.0

2.

90

3.14

5.

18

9.2

0.7

-0.3

44

0.00

0.

11

1.60

2.

13

0.0

-0.1

-0

.I

0.66

2.

24

4.24

0.

5 -0

.1

-0.1

0.

25

0.10

1.

38

1.85

0.

0 -0

.2

-0.1

0.

65

2.24

4.

25

0.7

-0.1

-0

.1

0.50

0.

07

1.13

1.

51

0.0

-0.2

-0

.1

0.65

2.

25

4.25

1.

1 -0

.2

-0.1

0.22

2 7

0.00

1.

43

2.27

2.

70

1.9

-0.8

-1

.2

1.44

2.

95

4.98

6.

5 0.

1 -0

.4

0.25

1.

15

1.92

2.

30

3.4

-0.6

--

1.3

1.41

2.

96

4.99

9.

2 0.

1 -0

.7

0.50

0.

78

1.48

1.

80

5.4

-0.5

-1

.3

1.36

3.

00

5.07

14

.6

-0.3

-1

.3

24

0.00

0.

23

1.57

2.

11

0.0

-0.2

-0

.1

0.33

2.

21

4.21

0.

9 -0

.2

-0.1

0.

25

0.19

1.

36

1.83

0.

1 -0

.2

-0.1

0.

33

2.21

4.

21

1.2

-0.2

-0

.1

0.50

0.

15

1.11

1.

49

0.1

-0.4

-0

.2

0.33

2.

21

4.22

2.

1 -0

.3

-0.2

0.66

7 3

0.00

1.

06

1.92

2.

40

4.6

-1.1

-1

.4

0.71

2.

71

4.75

13

.1

-1.0

-1

.4

0.25

0.

81

1.6

0

2.03

6.

3 -1

.0

-1.4

0.

69

2.78

4.

87

18.5

-1

.7

-2.1

0.

50

0.51

1.

24

1.60

7.

3 -1

.2

-1.4

0.

64

3.05

5.

41

24.3

-2

.7

-2.9

12

0.00

0.

28

1.51

2.

07

0.1

-0.2

-0

.1

0.15

2.

14

4.14

1.

7 -0

.2

-0.I

0.

25

0.24

1.

31

1.79

0.

2 -0

.3

-0.2

0.

15

2.14

4.

14

2.4

-0.3

-0

.2

0.50

0.

18

1.07

1.

46

0.5

-0.5

-0

.3

0.15

2.

15

4.15

4.

0 -0

.5

-0.3

2.00

0 3

0.00

0.

48

1.54

2.

10

1.9

-0.9

-0

.6

0.18

2.

21

4.22

8.

2 -1

.2

-0.7

0.

25

0.38

1.

33

1.81

3.

4 -1

.0

-0.7

0.

18

2.26

4.

30

12.1

-1

,7

-1.1

0.

50

0.26

1.

07

1.47

5.

4 -1

.0

-0.6

0.

17

2.53

4.

83

18.6

-2

.2

-1.4

8 0.

00

0.23

1.

46

2.03

0.

2 -0

.2

-0.1

0.

07

2.07

4.

07

2.4

-0.2

-0

.1

0.25

0.

19

1.26

1.

76

0.4

-0.2

-0

.1

0.07

2.

07

4.07

3.

4 -0

.3

-0.1

0.

50

0.14

1.

03

1.44

1.

1 -0

.3

-0.2

0.

07

2.08

4.

09

5.8

-0.4

-0

.2

z o t'-

z o z taO

Page 20: Calculating parameters for infiltration equations from soil hydraulic functions

Tab

le V

I. V

alue

s of

/3 a

nd (r

wit

h er

rors

of

anal

ytic

al a

ppro

xim

atio

ns fo

r B

rook

s an

d C

orey

wat

er r

eten

tion

func

tion

4~

A

fl

Err

or

a E

rror

rl

O

o ~

1=

0.9

h

i=0

h

l=l

O1

=0

.9

h~

=0

h

l=l

O1

=0

.9

hi=

0

hl=

l O

1=

0.9

h

i=0

h

1=

1

0.10

5 13

0.

00

1.00

1.

00

1.00

0.

00

0.00

0.

00

0.00

0.

00

0.15

0.

00

0.00

0.

02

0.25

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.00

0.

17

0.00

0.

00

0.02

0.

50

1.00

1.

00

1.00

0.

00

0.00

0.

00

0.00

0.

04

0.26

0.

00

0.01

0.

02

44

0.00

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.76

0.

87

0.00

0.

00

0.00

0.

25

1.00

1.

00

1.00

0.

00

0.00

0.

00

0.00

0.

76

0.87

0.

00

0.00

0.

00

0.50

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.76

0.

87

0.00

0.

00

0.00

0.22

2 7

0.00

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.14

0.

37

0.00

0.

02

0.03

0.

25

1.00

1.

00

1.00

0.

00

0.00

0.

00

0.00

0.

20

0.42

0.

00

0.02

0.

03

0.50

0.

99

0.99

0.

99

-0.0

1

0.00

0.

00

0.00

0.

35

0.54

0.

00

0.01

0.

02

24

0.00

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.80

0.

89

0.00

0.

00

0.00

0.

25

1.00

1.

00

1.00

0.

00

0.00

0.

00

0.00

0.

80

0.89

0.

00

0.00

0.

00

0.50

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.80

0.

89

0.00

0.

01

0.00

0.66

7 3

0.00

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.47

0.

65

0.00

0.

02

0.02

0.

25

0.96

0.

93

0.92

-0

.02

0

.01

0

.02

0.

00

0.56

0.

71

0.00

0.

01

0.02

0.

50

0.83

0.

75

0.71

-0

.08

0.

03

0.07

0.

00

0.66

0.

78

0.00

0.

01

0.01

12

0.00

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

00

0.87

0.

93

0.00

0.

00

0.00

0.

25

1.00

1.

00

1.00

0.

00

0.00

0.

00

0.00

0.

87

0.93

0.

00

0.01

0.

00

0.50

0.

99

0.99

0.

99

0.00

0.

00

0.00

0.

00

0.88

0.

94

0.03

0.

01

0.00

2.00

0 3

0.00

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

04

0.85

0.

92

0.04

0.

01

0.01

0.25

0.

90

0.84

0.

83

-0.0

1

0.05

0.

06

0.02

0.

86

0.92

0.

04

0.02

0.

01

0.50

0.

78

0.62

0.

59

-0.0

7

0.12

0.

14

0.02

0.

87

0.93

0.

03

0.02

0.

01

8 0.

00

1.00

1.

00

1.00

0.

00

0.00

0.

00

0.01

0.

94

0.97

0.

02

0.00

0.

00

0.25

1.

00

1.00

1.

00

0.00

0.

00

0.00

0.

03

0.94

0.

97

0.03

0.

00

0.00

0.

50

0.94

0.

94

0.94

0.

00

0.02

0.

02

0.03

0.

94

0.97

0.

04

0.01

0.

00

5~

Page 21: Calculating parameters for infiltration equations from soil hydraulic functions

INFILTRATION EQUATIONS FROM SOIL HYDRAULIC FUNCTIONS 335

5 I*

I

5 i0

ol i

o 5 IO

t * t *

z 2 . 5 .

5

o.

(c)

S i

0.7 0.9

0

0 .... I

(d)

zlO

20 I

.5 0.7

0

0.9

fo.5

0 0.5

i

0.7 0.9

fO.5

0 0.5 0.7 .9

O O

Figure 5. Dimensionless infiltration I* plotted against dimensionless time t* = 2t/tgrav, profile of normalised water content O at t* = I0 against dimensionless depth z, and flux- concentration relation f(O). Figures for Grenoble sand and Yolo light clay are on the left and right respectively. Solid lines for I* and O were obtained from numerical solution of Richards' equation, while the solid line for f(O) was obtained by numerical integration of the Brace and Klute (1956) equation. Dashed lines in (e) and (f) show the approximation f(O) ~ 2(0 - 0 0 ) / ( 0 + O1 - 200) used for approximate infiltration parameters. Dashed lines in (a) and (b) obtained from the infiltration equation using both accurate and approximate infiltration parameters are very close to the lines from the numerical solution.

5. Conc lus ions

We have presented d imens ion less implici t analyt ical equat ions for infiltration o f

water at a negat ive o r posi t ive head into a h o m o g e n e o u s soil at un i fo rm initial water content . The equat ions involve d imens ionless parameters that can be ca lcula ted

Page 22: Calculating parameters for infiltration equations from soil hydraulic functions

336 P.J . R O S S E T AL.

0 Uo.

M

, i

- 1 0 10

2

i

0

- 1 0

. . . . . . . . . . . .

s ,S

5 10

t *

2

N

- 1

0 5 10

( : t

ot M

- 1

0

t

(d)

,r

i

5 10

t::*

N

go I.i 1,1

-1

I

(e)

i I

5 10

2 ( f )

i

io N

- 1 �9 , i

0 5 10

Figure 6. Percent errors in infiltration as a function of dimensionless time t* = 2t/tgrav calculated for Grenoble sand using accurate (full lines) and approximate analytical (broken lines) infiltration parameters. (a): Oo = 0, O1 = 0.9; (b): Oo = 0:5, O1 = 0.9; (c): Oo = 0, O1 = 1,hz = 0; (d): Oo = 0.5, O1 = 1,h~ = 0; (e): Oo = 0, Ol = 1,hi = 1; (f) Oo = 0.5, O1 = 1,hi = 1.

from the normalised soil water retention and hydraulic conductivity functions. Analytical approximations to the parameters are given for both the van Genuchten and the Brooks and Corey water retention functions combined with the Brooks and Corey hydraulic conductivity-water content relation. Application of the equations involves the parameters used to normalise thehydraulic functions, which are a head scaling factor, the saturated and residual water contents, and the saturated hydraulic conductivity; but these enter very simply into the calculation of infiltration, time, and initial and boundary conditions as given by Eqs (2) and (5).

Page 23: Calculating parameters for infiltration equations from soil hydraulic functions

INFILTRATION EQUATIONS FROM SOIL HYDRAULIC FUNCTIONS 337

• 2

$I

M

-2

I

_.. (a)

I

5 10

N

-2 o

I

(b)

I

5 10

t:*

4

A

A* 2.

o" ~o.

-2

I

(c)

5 i0

4

0

-2

I

(d)

5 i0

14 0

N

,I

(e)

I

5 10

M

-21 0

l '(f) '

J

5 10

t "

Figure 7. Percent errors in infiltration as a function of dimensionless time t* = 2t/tgrav calculated for Yolo light clay using accurate (full lines) and approximate analytical (broken lines) infiltration parameters. (a): 6)0 = 0, 6)1 = 0.9; (b): 6)o = 0.5, O1 = 0.9; (c): 6)o = 0,6)1 = 1,hi = 0; (d): 6)o = 0.5,6)1 = 1,hi = 0; (e): 6)0 = 0,6)1 = 1,hi = 1; (f) (90 = 0.5, (91 = 1, hi = 1.

T h e equa t ions desc r ibed infil tration into a sand and a c lay o v e r a r ange o f initial and b o u n d a r y condi t ions and t imes with an er ror o f less than three percent . The ana ly t ica l a p p r o x i m a t i o n s to the infi l tration p a r a m e t e r s were accura te for a wide r ange o f hydrau l i c p a r a m e t e r s r ep resen t ing the major i ty o f soil types , a l though a c c u r a c y de te r io ra ted for re la t ively large values o f the van G e n u c h t e n m p a r a m e t e r c o m b i n e d wi th a smal l r/, where r / i s the p o w e r in the Brooks and C o r e y hydrau l ic conduc t iv i ty funct ion.

Page 24: Calculating parameters for infiltration equations from soil hydraulic functions

338 P.J. ROSS ET AL.

Whi l e the equat ions involve two paramete r s , fl and a , in addi t ion to sorpt iv i ty and conduc t iv i ty , /3 is usua l ly c lose to 1 for dry initial condi t ions . T h e p a r a m e t e r a takes into accoun t the near -sa tura ted par t o f the wa te r re tent ion curve , and is

the re fore par t icular ly impor t an t for infil tration expe r imen t s p e r f o r m e d at a sl ight

suct ion.

Acknowledgements

T h e senior au thor a c k n o w l e d g e s 'Cen t r e Na t iona l de R e c h e r c h e Sc ien t i f ique ' ( C N R S ) o f F r ance for f inancial suppor t dur ing d e v e l o p m e n t o f this work .

References

Barry, D. A., Parlange, J.-Y., Haverkamp, R. and Ross, E J.: 1995, Infiltration under ponded conditions: 4. An explicit predictive infiltration formula, Soil Sci. 160, 8-17.

Brooks, R. H. and Corey, T. H.: 1964, Hydraulic properties of porous media. Hydrology paper 3, Colorado State Univ., Fort Collins, Colo.

Bruce, R. R. and Klute, A.: 1956, The measurement of soil-water diffusivity, Soil Sci. Soc. Am. Proc. 20, 458--462.

Fuentes, C., Haverkamp, R. and Parlange, J.-Y.: 1992, Parameter constraints on closed-form soilwater relationships, J. Hydrol. 134, 117-142.

Haverkamp, R., Vauclin, M., Touma, Y., Wierenga, P. and Vachaud, G.: 1977, A comparison of numerical simulation models for one-dimensional infiltration, Soil Sci. Soc. Amer. J. 41, 285- 294.

Haverkamp, R., Parlange, J.-Y., Starr, J. U, Schmitz, G. and Fuentes, C.: 1990, Infiltration under ponded conditions: 3. A predictive equation based on physical parameters, Soil Sci. 149, 292- 300.

Haverkamp, R., Fuentes, C. and Parlange, J.-Y.: 1992, Toward a universal choice of soil hydraulic properties: closed-form relations and/or integral parameters? in: M. th. Van Genuchten and E J. Leij. Indirect Methods for Estimating the Hydraulic Properties of Unsaturated Soils, (eds), pp 213-217. University of California, Riverside.

Haverkamp, R., Ross, P. J., Smettem, K. R. J, and Parlange, J.-Y. 1994, Three-dimensional analysis of infiltration from the disc infiltrometer: 2. Physically based infiltration equation, Water Resour. Res. 30, 2931-2935.

Haverkamp, R., Ross, P. J., Parlange, J.-Y., Tranckner, J. and Bohne, K.: 1995, Infiltration under ponded conditions: 5. Prediction of infiltration parameters with changing initial condition, Soil Sci. (submitted).

Moore, R. E.: 1939, Water conduction from shallow water tables, Hilgardia 12, 383-426. Parlange, J.-Y.: 1971, Theory of water movement in soils: 1. One-dimensional absorption, Soil Sci.

111, 134--137. Parlange, J.-Y.: 1975, On solving the flow equation in unsaturated soils by optimization: Horizontal

Infiltration, Soil Sci. 133, 337-341. Parlange, J.-Y., Lisle, I., Braddock, R.D. and Smith, R. E.: 1982, The three-parameter infiltration

equation, Soil Sci. 133, 337-341. Parlange, J.-Y., Haverkamp, R. and Touma, J.: 1985, Infiltration under ponded conditions: 1. Optimal

analytical solution and comparison with experimental observations, Soil Sci. 139, 305-311. Philip, J. R.: 1957, The theory of infiltration: 5. The influence of the initial moisture content, Soil Sci.

84, 329-339. Philip, J. R.: 1969, Theory of infiltration, Adv. Hydrosci. 5, 215-305. Philip, ]. R.: 1973, On solving the unsaturated flow equation: 1. The flux-concentration relation, Soil

Sci. 116, 328-335. Ross, P. J. and Bristow, K. L.: 1990, Simulating water movement in layered and gradational soils

using the Kirchhoff transform, Soil Sci. Soc. Am. J. 54, 1519-1524.

Page 25: Calculating parameters for infiltration equations from soil hydraulic functions

INFILTRATION EQUATIONS FROM SOIL HYDRAULIC FUNCTIONS 339

Smettem, K. R. J., Bristow, K. L., Ross, P. J., Haverkamp, R., Cook, S. E. and Johnson, A. K. L.: 1994, Trends in water balance modelling at field scale using Richards' equation, Trends in Hydrology 1, 383-402.

Touma, J., Vachaud, G. and Parlange, J.-Y.: 1984, Air and water flow in a sealed ponded vertical soil column, Soil Sci. 137, 181-187.

Van Genuchten, M. Th.: 1980, A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J. 44, 892-898.

Wolfram Research, Inc.: 1992, Mathematica, Wolfram Research, Inc., Champaign, Illinois.