Journal of Hydrology, 142 (1993) 29-46 Elsevier Science Publishers B.V., Amsterdam

[31

A class of exact solutions for Richards'

29

equation

D . A . B a r r y a, J . -Y. P a r l a n g e a.~, G . C . S a n d e r b a n d M. S ivap lan a

aCentre for Water Research, University of Western Australia, Nedlands, W.A. 6009, Australia bFacully of Science and Technology, Griffith University, Nathan, Qid. 4111. Australia

(Received 1 May 1992; revision accepted 11 July 1992)

ABSTRACT

Barry, D.A., Parlange, J.-Y., Sander, G.C. and Siva#an, M., 1993. A class ofexact solutions for Richards' equation. J. Hydrol., 142: 29--46.

A new solution satisfying Richards' equation is derived. The solution, which may be applied for infiltration or capillary rise, is valid for the condition of an arbitrary moisture tension imposed at the soil surface. When written in terms of the moisture tension, the new result is very simple, being derived in terms of a similarity variable. The solution applies when the form of the soil moisture characteristic curve is a particular weighted integral of the gradient of the unsaturated hydraulic conductivity. Thus, if the soil moisture characteristic curve is selected a priori, then this condition determines the hydraulic conductivity. The converse of this statement also applies. The cumulative infiltration derived from the solution is of the form of the Green-Ampt infiltration equation; however, there is no need to assume a steep wetting front as Green and Ampt did. Finally, using the correspondence between Richards' equation and the convection- dispersion equation with a non-linear solute adsorption isotherm, a new exact solution for adsorptive solute transport is d~,!ved.

I N T R O D U C T I O N

Exact solutions for water movement in soil have wide theoretical and practical applicability. The governing unsaturated flow equation derived by Richards (1931) is highly non-linear and, consequently, difficult to solve exactly in closed form. Exact solutions, when they can be found, may be used directly or as realistic checks of numerical schemes. For these reasons, an apparently new solution for Richards' equation under the condition of a fixed head surface boundary condition is derived. This boundary condition has significant importance in practice. Unlike some recent solutions for Richards' equation (e.g. Broadbridge and White, 1988; Sander et al., 1988; Barry and

Correspondence to: D.A. Barry, Centre for Water Research, University of Western Australia, Nedlands, W.A. 6009, Australia.

Permanent Address: Department o f Agricultural and Biological Engineering, Cornell Uni- versity, Ithaca, NY 14853-5701, USA.

0022-1694/93/$06.00 © 1993 - - Elsevier Science Publishers B.V. All rights reserved

30

N O T A T I O N

D.A. B A R R Y ET A L

a l A A, B C

Co C!

CB

D

f g I K Ks li N, P q S

so S t

V

W 2

• 7 I

,Tf

constant, L function of time i -- 1, 2, 3. arbitrary constants constant, L normalised solute concentration in the aqueous phase normalised influent solute concentration arbitrary function of time fitting parameter for the solute adsorption isotherm given in the text soilwater diffusivity, L2T - dispersion coefficient, L2T - an unknown function of z, L magnitude of the gravitational acceleration [LT -z] cumulative infiltration, L hydraulic conductivity, LT- t hydraulic conductivity at complete saturation, LT- logarithmic integral i = 1, 2, constants defined by (A3) and (A4) fluid pressure offset so that p = 0 at atmospheric pressure [ML-~T -2] Darcy flux, LT--i normalised solid phase solutc concentration normalised solid phase solute concentration at c = Co sorptivity, LT- ~/2 time, T mean solute velocity, LT- transcendental function defined in the text depth below the soil surface [L] an arbitrary position in the soil profile, L position of the solute front, L

Greek letters

0t

~t*

0 ®

0F 0~ 0surface

2

P 4,

qJa

qJs ff

constant constant, T volumetric moisture content normalised moisture content residual moisture content volumetric moisture content at complete saturation moisture content at the surface similarity variable, LT-~/2 density of the soil water, ML -3 similarity variable, L moisture tension, L air-entry moisture tension, L moisture tension at the soil surface, L dimensionless variable defined in the text by (26)

Sander, 1991; Rogers et al., 1983), the new solution does not depend on any specific, algebraic form of the soil hydraulic functions. Rather, there is a general functional dependence between the soil moisture characteristic curve and the gradient of the soil hydraulic conductivity.

RICHARDS" EQUATION 31

The solution is analysed to determine its suitability for practical applica- tions. One application is the prediction of infiltration rates, or cumulative infiltration. Many algebraic infiltration laws have been presented in previous hydrological literature. One, in particular, derived by Green and Ampt (1911 ), has been the basis of numerous hydrological studies and was developed as a natural consequence of the solution. However, unlike the Green-Ampt solution, the re-derived model does not rely on the assumption of a sharp wetting front.

Many authors (e.g. Lantz, 1970) have noted the connection between Richards' equation and the transport equation for a solute that is being partitioned into the liquid and solid phases, where the partitioning is described by an equilibrium adsorption isotherm. Thus, finding a new solution for Richards' equation leads directly to a new exact solution for adsorptive solute transport.

EXACT SOLUTION OF RICHARDS' EQUATION

The governing equation for water movement in unsaturated soil derived by Richards (1931) can be written with water content or moisture tension as the dependent variable. In terms of the moisture tension, • = p(pg)-J, the one- dimensional form of Richards' equation is

dOdq ~ c3[ dq j ] dW dt = Oz K(W) t3z K(W) (1)

For convenience, all symbols are defined in the notation list. Equation (1), in which z is taken as positive downwards from the soil surface, describes isothermal movement of incompressible soil water in a structurally rigid porous medium. Soil air movement is assumed to have a negligible effect on the water flow. The soil water is taken to be homogeneous, as is the hydraulic conductivity, K. Further, the soil moisture characteristic curve, 0(~) or ~(0), is assumed to be single valued, i.e. hysteresis is ignored. This latter restriction is unimportant since interest is primarily in infiltration and capillary rise.

Solution of the infiltration model

The key step in finding a solution satisfying (1) is the functional form assumed for the moisture content. For many soils there is an effective air-entry tension, W, (Haverkamp et al., 1990). Below this value, the water content is less than 0 s, the moisture content at complete saturation. We consider soils

32 D.A. B A R R Y ET AL.

whose soil moisture characteristic curve has the form:

l <

0(~P)- 0, ~ ( V ' + B) dW' (2) o ( ~ ) = o ~ - or = -

1 ~ > ~ .

where ~ is an arbitrary constant and 0r is the residual water content (which could be zero). An appropriate value for • is obtained by requiring that O ~ 1 as W -~ W,, thus

~u a

= K~ tp~ _ , (tp + B) dW (3)

It is clear that (2) applies to any soil moisture characteristic curve and unsaturated hydraulic conductivity function, with the proviso that only one of these functions can be chosen independently. If, for example, the soil moisture characteristic curve is known, (2) can be used to find K such that the exact solution given below holds. The physical implications of this relation- ship are discussed below (22).

The solution presented below is also valid for a surface potential that is less than W,. For such a (constant) value assigned to ~(0, t), there will correspond 0.~,rr, c~ < 0~. In this case, the solution applies with 0~ in (2) replaced by 0surr, c~ and q-', in the integral in (3) replaced by W(0, t).

Using (2), the soil water capacity is:

{(0 - ~ * dK(W) dO = ' ~ B) ~ ~' < ~''' dq j

where the constant, 0~* ( > 0), is defined by:

(4)

~, = -(0~ - 0r)~q'~ K~ (5)

Equation (2) indicates that for a saturated surface there are two parts of the solution to consider: (I)q' ~> W,,(® = 1) and (II)q' < qJ.,(® < 1). Both parts of the solution are examined below.

Part I: 7' >1 tp,. The left-hand side of (1) vanishes and, since K = K~, the right side simplifies considerably to

82~ 0 = 8z 2 (6)

RICHARDS" EQUATION 33

The general solution of (6) has the well-known dependence on z

~F(z, t) -- q'~ + cl (t)z

where ~ is the value of W at z = 0.

(7)

Part II: 'P < tP a. Substitution from eqn. (4) into the left-hand side of eqn. (1) and differentiating the bracketed term on the right gives

- a * d[ln K(W)] t~V d[lnK(V)] OV (OW ) c32q' (W + B) dW Ot = dW Oz . -~z 1. + c~z~ (8)

qJ(z, t) satisfying eqn. (8) is sought. Motivated by the similarity methods used in finding solutions of the absorption equation, let W(z, t) = W(~b), where 4~ = f (z ) lA( t ) . Equation (8) may now be rewritten as:

A(t)q~* d[lnK(V)] dV dA(t) d/(z) drinK(V)] dV [dV d/(z) A(t) ] (~ + B) d ~ dq~ dt = dz d ~ d~ [ d ~ (iz J

d2V [ df(z) ]2 d~ d2f(z) +-d--~[_ dz _] + A(t) d~ dz 2 (9)

(9) immediately suggests as a solution, ~ - ~ - B and + z. With these assumptions, eqn. (9) simplifies considerably to:

E q u a t i o n

f ( z ) = a,

dA ~*A - 1 - A (10)

dt

The function A(t) is determined by (10). The general solution to (10) is:

-t~, = A ( t ) - A(O) + In ! - A(O) (11)

In general, A(t) must be found numerically from (l l). The choice of A(O) will subsequently become evident when the surface boundary condition is considered.

Collecting the results from the foregoing analysis, the solution of • < ~P, is given by the expression:

= aj + z B (12) A(t)

It should be noted that the flux of water being transmitted through the unsaturated portion of the soil profile, q(z, t), is from both Darcy's law and (12):

q(z, t) 1 - 1 ( 1 3 )

g[V(z, t)] ,4(0

34 D.A. BARRY ET AL

Solution valid for all ~' The complete solution is obtained by requiring continuity oftP at W = W,.

Thus, eqns. (7) and (12) are equated at W = W,; c~(t) is found to be the inverse of A(t) and W~ is a time-depende,,.t function given by:

Ws(t ) = al B (14) A(t)

Equation (14) represents the functional form of the fluid head at the surface boundary. It should be noted that it includes the case of a constant Ws, a case of significant practical importance. If W~ is time-dependent, then the time dependency must be inversely proportional to A(t) for the present solution to be valid. Thus, the complete solution, valid for all • ~< ~ , is only (12).

Application for q'~ constant If the condition that W is fixed to q~ at ~, - ,.I for all t is imposed, then

solution (14) applies with a~ = - z l and B = -W~. By a simple translation of the position variable, z, z~ can always be moved to position 0 in the new coordinate system. The complete solution, with a~ = 0, is then written as:

,7. = + (15)

A(t)

Observe that the flux calculated in (13) for the unsaturated zone applies also in the saturated zone. The general solution (12) depends on three parameters: A(0), B and a~. For a fixed qJ~ at z = 0 (15) shows that B and a~ are determined. Below, it is shown that A(0) is also easily determined.

Illustration." #!filtration hlto a do' soil under constant potential

In this case ~ is a constant, thus if W, > W,, the surface is saturated but not ponded unless tlJ~ > 0. Equation (14) indicates that at = 0. It also indicates that A(0) = 0, thus the flux is infinite at time zero. With A(0) now specified, it is useful to return to the definition of A(t) given by (11). An explicit expression for A(t) is:

[ (-')1 A(t) = 1 + w - e x p ~ 1 (16)

where the function W(x) is defined implicitly by the equation:

x = W(x) exp [W(x)] (17)

According to Fritsch et al. (1973), this function was first considered by L. Euler. This function rather than (11) is used because it is easy to manipulate

R I C H A R D S " E Q U A T I O N 35

0 f _, cooi,,o .ise

x

~= - 3 Infiltrotion

- 4

- 5

- 6 i , ,

-0 .5 0.0 0.5

=

1.0 1.5 2.0

Fig. i. The W function defined by (17). It should be noted that dW(x)/dx is infinite at x = - e x p ( - I). As discussed in the text, the relevant portion of the W function lies in the region x = ~< 0, where the W function is double valued. The upper port ion of the curve ( - I <~ W ~< 0) applies in the case of capillary rise. The lower portion (W ~< - 1) applies for infiltration.

formally, e.g. differentiation and integration of the W function can be carried out. What is more important, however, is that the behaviour of the solution is easily pictured in terms of the W function, as depicted in Fig. 1. Equation (16) and Fig. 1 make clear the behaviour of the solution (15), a much easier task than trying to visualise the function A(t), in (! 5), from the implicit eqn. (1 l). It should be noted, that for any x in the range - exp ( - l) < x ~< 0, the W function is double valued. It is apparent from (16) that, since ~* > 0, it is precisely this range that is of interest here. Indeed, for infiltration, the appro- priate branch of the W function is the lower one, i.e. W ~< - 1. For capillary rise, the useful branch would correspond to - l ~< W < 0. The implicit definition of the W function given by (17) is a little inconvenient for computa- tional purposes. For this l cason, simple, explicit approximations for both branches of the W function in the range - e x p ( - 1) ~< x < 0 are included in the Appendix.

For q~ > ~F.,, K[~P(0, t)] = K~ and (10) and (13) yield:

I(t) = - ~ * K s A ( t ) (18)

where l ( t ) is the cumulative infiltration given by: l

= f q(0, t) dT (19) l(t) a 0

The short time expansion of (18) defines the sorptivity of this class of soils as

36 D.A. BARRY ET AL.

S = Ks ~/2~*. Further, (11) and (18) yield:

Kst = / - ~ * K f l n [ l + ~ - ~ s ] (20)

which has the general form of the Green and Ampt (1911) law without requiring a sharp wetting front. The model presented by Green and Ampt (1911) is widely used to predict infiltration into dry soils. It has also been modified and used as the underpinning of many other infiltration equations (e.g. Hillel and Gardner, 1970; Mein and Larsen, 1973: Chu, 1978; Freyberg et al., 1980; Davidson, 1984, 1987; Reid and Dreiss, 1990).

The soil water profile is deduced from (4) and (15) as

z _ ~ , d K = d--ff ~P ~< W, (21)

which clearly implies that

d2K < 0 (22) d02

As pointed out by Milly (1985), most natural soils satisfy the opposite condition to (22), i.e. d2K(O)/dO 2 > 0. Thus, 'soils' corresponding to (22) occur very infrequently, if at all. Therefore (2) cannot be considered as a realistic relationship between hydraulic conductivity and the soil moisture characteristic curve, as there is a variety of more useful models available (e.g. Burdine, 1953; Mualem, 1976). On the other hand, infiltration into dry soils is dominated by the soil water diffusivity close to the maximum water content in the soil. Near this maximum, air entrapment typically causes the local curvature of the unsaturated hydraulic conductivity to become negative (McWhorter, 1971; Parlange and Hill, 1979). It is interesting to recall that Green and Ampt (1911) investigated both capillary rise and infiltration. They found their infiltration law fitted, with reasonable precision, data from their infiltration experiments. In contrast, data from their capillary rise experiments could not be fitted very well. These results can be explained by (22). For infiltration, air entrapment is to be expected; this means that (22) holds for higher values of 0 and thus the Green and Ampt model will be a reasonable approximation. On the other hand, for capillary rise, very little air will be trapped and (22) will never be satisfied.

To illustrate the form of the soil moisture profiles consider the ad hoc example:

" " [ ] r..<,-,.] W~ = 1 - 7 1 + t l j (~_--O,) cosL2(02 ~/) (23)

RICHARDS" EQUATION 37

Normalized Water Content, ® 0,0 0.5 1.0

0

- 2

£ o

P-- - 4 u~ o

O_

"o - - 6 N

O Z

- 1 0

t/a" = 2

t / ¢ " = 5

f

Fig. 2. Dimensionless water content profiles calculated from (25) with ( = - I.I.

Clearly, this equation applies in the unsaturated part of the profile. For simplicity, taking Ws = 0, eqn. (4) gives (for 0 < 0s)

[ Wa(0s -- 0 r ) ] s in [ it(0 - 0r) ] Wa(0 -- 0r) (24) K = Ks + ~x* 2(0~ - Or) - ~x*

The unsaturated part of the profile for this case is found from (21) as

= ~" 2 ( 0 ~ - 0,) + ~ - cos 2(0~- 0r) J

Some normalised profiles from (25) are given in Figs. 2 and 3. The difference

N

c - 1 0 o

0~ o

12.

-o

.___N --20 t~

E o

Z

- 3 0

Normalized Water Content, ® 0.0 0,5 1.0

0 i

Fig. 3. Same as Fig. 2 except that ~ = - 3.

38 D.A. BARRY ET AL.

between these figures is in terms of the dimensionless variable, (, defined by:

0C'Ks (26) = 0r) ]

It should be noted that ~ = - 1 corresponds to a Green-Ampt profile, else < - 1. In Fig. 2 ~ = - 1.1, which is quite close to the Green-Ampt case.

The steep fronts in the water content profiles confirm this. On the other hand, in Fig. 3 ~ -- - 3. The profiles are quite diffused in this case. The position of the saturated front is identical in each figure for equal values of t/~*. This result follows after setting tI-' = tg,~ in (15) and solving for z, In both Figs. 2 and 3, the profiles " look" realistic. In fact they approximate the true profile in much the same way as the Brooks and Corey (1964, 1966) soil moisture characteristic curve approximates the matric potential. Also, (18) indicates that, for this class of soils, the quantity of water in the profile differs only by a constant for all t[o~*. Finally, if tp, = 0, the profile would not look so realistic (Parlange (1980) considered that case briefly).

Call Az the distance between the saturated and wetting fronts, so that:

Az -~x*K~ [1 + 1] = 2 ( 0 s - 0r) ( (27)

Equation (27) confirms that ~ ~< - 1. When the front is a step function, (27) yields ~ = - 1, or

~*K~ = - tP,,,(0, - 0r) (228)

Green and Ampt (1911) effectively define tp,, by (28). Obviously this unneces- sary requirement affects the shape of the profiles dramatically but not, however, the infiltration law that was the only concern of Green and Ampt.

Connection with the Bruce and Klute equation

Bruce and Klute (1956) used the Boltzmann similarity variable, 2 = z/x/~, to reduce Richards' equation (without gravity) to

1 d2 'i 2 dO (29) D(O) = 2 dO Or

where the initial moisture content in the soil is taken to be 0r: Bruce and Klute (1956) used (29) to determine the diffusivity from water content profiles. For the present case of Richards' equation with gravity:

1 i D(O) - g* dO ~b dO (30) Or

RICHARDS" EQUATION 39

where tk = zlA(t). From the short time limit in which A(t) approaches x/~/~* it is clear that, in this limit (30) reduces to (29), as expected.

A new solution for adsorptive solute transport

The correspondence equation and

~[s(e) + c] CO2e cot = D~ COZ2

which describes solute

between the moisture content form of Richards'

COc M _ v d--z ( 3 1 )

transport subject to an equilibrium adsorption isotherm, has been discussed by various authors including Barnes (1986, 1989), Yortsos (1987) and Barry and Sander (1991). It should be noted that in (31) the usual meanings of the symbols are used, i.e. s is the sorbed solute concentration, Ds is the solute dispersion coefficient and v is the mean pore water velocity (assumed constant). It has also been demonstrated that (31) maps directly onto (1) via the transformations (Barry et al., 1991):

v~ =

A2K(~)

and

Ai + A20

D~ ln(c + AzA3) (32)

= v(c + A2A3) (33)

= c + s(c) (34)

Equations (32) and (33) define K to be

K(~) - A2 exp • (35)

i.e. the Gardner (1958) formulation of the unsaturated hydraulic conductivity for constant v. Equations (32) and (33) fix K to be that given by (35); no other choice is possible. One use of (32) through (35) is to derive new solutions for (31) from solutions of Richards' equation, or vice versa.

For the purpose of illustration (31) is to be solved for a soil column experiment where solute at a known concentration Co is added to the column. For convenience, the solute concentration has been normalised by a suitable constant, so that dimensionless concentrations are dealt with. Initially, the solute is not present in either the solid or liquid phases. The initial and boundary conditions which are to be satisfied for the liquid phase concen- tration are:

c(O, t) = Co (36)

c(z, O) = 0 (37)

40 D.A, BARRY ET AL.

and

c(oo, t) = 0 (38)

These conditions are similar to those often applied in soil column experiments, e.g. the cation exchange experiment performed by Schweich et al. (1983).

It is necessary to relate the various quantities in the solute transport problem to corresponding quantities in the unsaturated flow solution. Thus (34) indicates that 0 is a function of c, i.e. 0 = O(c). The initial condition (37) indicates that A3 = 0 and A, = 0 from (32) and (34), respectively. Since the influent solution is at concentration Co, corresponding to ~ , (32) gives Co = exp(v~..,/Ds). Corresponding to ~s, there is a "concentrat ion" CB = exp(vqJ~/Ds), which is shown below to be a fitting parameter. Substitut- ing these results into (2) gives:

0 ( 0 = -li(c~) (39, where li is the logarithmic integral (Spanier and Oldham, 1987, Section 37 : 2,. Of course, the function O(c) is only the mathematical relationship (34); it does not mean that the water content is a function of solute concentration. To obtain (39), assume 0s = O(co), which simply removes an unnecessary constant. At e = Co the total quantity of solute adsorbed by the solid phase is s(c0), thus from (34) and (39):

Co + S( Co ) A2 = (40) ,i(c=:) The constant, e*, is then given by (5) as:

~, = -D~[co + s(co)l

V2CB li ( ~ ) (41)

The combination of (34), (39) and (40) define the adsorption isotherm, s(c). This isotherm is parametrized by the fitting parameter CB. The complete solution consists of two parts:

c = CBexp D~A(t) z > zf

Co otherwise

The position of the front, zr, is defined by:

zf = " In V

(42)

(43)

RICHAROS" EQUATION 41

0

f~

1.00

0.75

0.50

0,25

0,00 -

-0 .25 0.0 0.5 1.0

c/co Fig. 4. Normalised adsorption isotherm calculated using (34), (39) and (40). The parameter values are given in the text.

The physical interpretation of this solution is demonstrated by way of a numeric example. For this purpose, set cB = 5c0 and S(Co) = So = co. The adsorption isotherm for this case is displayed in Fig. 4. In this figure, it is apparent that the isotherm becomes negative for low values of C[Co. Although this effect becomes negligible for larger values of cB, it does not disappear. However, apart frora this defect, the isotherm looks quite realistic, and is reminiscent of the lower portion of an "S-curve" isotherm, according to the classification scheme reported by Sposito (1989, fig. 8.1).

Figure 5 contains two profiles calculated using (42). As with the water

1.0

0 (J

~ 0 . 5 0

0.0 0

i

5 10 15

vz/D, Fig. 5. Dimensionless concentration profiles for the isotherm in Fig. 4.

42 D.A. BARRY ET AL.

content profiles of Figs. 2 and 3, the sharp corner at z = zf is unrealistic. It exists because the concentration has been set at c = Co for z < zf. For z > zf, the profiles are quite dispersed, as one would expect for the type of isotherm shown in Fig. 4.

The solution presented here, although extremely simple, appears to have two non-physical aspects. The first is rather mild in that the adsorption isotherm has a negative gradient at z = 0. Second, and more important, is the abrupt change in slope in the concentration profiles around z = zf. Because of this latter behaviour, it may be that the solution is more useful as a check on numerical schemes rather than one to be used in direct applications to experimental data.

CONCLUSIONS

Equation (15) has been presented as an extremely simple solution to Richards' equation, with fixed head boundary and initial conditions that is valid for the class of soils whose soil moisture characteristic curve can be represented by (2). The new solution is closely related to two fundamental results in unsaturated soil moisture flow. First, it is a natural extension of the approach adopted by Bruce and Klute (1956) who used the Boltzmann similarity variable to simplify the absorption equation. Second, the cumulative infiltration calculated for this class of soils is mathematically identical to the widely used Green-Ampt infiltration law. However, the approximation of a sharp wetting front is no longer required.

While the solution presented above is simple, it relies on soil hydraulic functions that vary with the head imposed at the surface boundary. This restriction means that the model can be used to replicate particular experiments. However, experiments carried out with the same soil but with different surface boundary conditions will not necessarily be predicted with the same model parameters.

The solution for infiltration leads also to a new exact solution for solute transport subject to a nonlinear equilibrium adsorption isotherm. Although the nature of the adsorption isotherm means that the solution has some physical relevance to modelling adsorptive solute transport, it is suggested that, like the corresponding solution for Richards' equation, its extremely simple form makes it more suitable for testing numerical schemes of analytical approximations.

ACKNOWLEDGEMENTS

Support for D.A. Barry was provided by the Australian Research Council

RICHARDS' EQUATION 43

and the Water Authority of Western Australia. J.-Y. Parlange acknowledges support from the University of Western Australia in the form of a Gledden Visiting Senior Fellowship.

REFERENCES

Barnes, C.J., 1986. Equivalent formulations for solute and water movement in soils. Water Resour. Res., 22(6): 913-918.

Barnes, C.J., 1989. Solute and water movement in unsaturated soils. Water Resour. Res., 25(!): 38-42.

Barry, D.A. and Sander, G.C., 1991. Exact solutions for water infiltration with an arbitrary surface flux or nonlinear solute adsorption. Water Resour. Res., 27(20): 2667-2680.

Barry, D.A., Sander, G.C. and Phillips, I.R., 1991. Modelling solute transport, chemical adsorption and cation exchange, Int. Hydrology and Water Resources Sym. 2-4 October, Perth, W.A. The Institution of Engineers, Australia, Nat. Conf. Publ. No. 92/19 (Preprints of Papers, Vol. 3: 924-929).

Broadbridge, P. and White, I., 1988. Constant rate rainfall infiltration: A versatile nonlinear model, !, Analytic solution. Water Resour. Res., 24(1): 145-154.

Brooks, R.H. and Corey, A.T., 1964. Hydraulic Properties of Porous Media. Hydrology Paper No. 3. Colorado State Univ., Fort Collins, CO.

Brooks, R.H. and Corey, A.T., 1966. Properties of porous media affecting fluid flow. J. lrrig. Drain. Div., Am. Soc. Civ. Eng., 92(IR2): 61-88.

Bruce, R.R. and Klute, A., 1956. The measurement of the soil moisture diffusivity. Soil Sci. Soc. Am. Proc., 20(4): 458-462.

Brutsaert, W., 1977. Vertical infiltration in dry soil. Water Resour. Res., 13(2): 363-368. Burdine, N.T., 1953. Relative permeability calculation from size distribution data. Trans.

AIME, 198: 71-78, Chu, S.T., 1978. Infiltration during an unsteady rain. Water Resour. Res., 14 (3): 461-466. Davidson, M.R., 1984. A Green-Ampt model of infiltration in a cracked soil. Water Resour.

Res., 20: 1685-1690. Davidson, M.R., 1987. Asymptotic infiltration into a soil which contains cracks or holes but

whose surface is otherwise impermeable. Transp. Porous Media, 2(2): 165-176. Freyberg, D.L., Reeder, J.W., Franzini, J.B. and Remson, J., 1980. Application of the Green-

Ampt model to infiltration under time-dependent surface water depths. Water Resour. Res., 16(3): 517-528.

Fritsch, F.N., Shafer, R.E. and Crowley, W.P., 1974. Solution of the transcendental equation w e " = x . Commun. Assoc. Comput. Mach., 16(2): 123-124.

Gardner, W.R., 1958. Some steady state solutions of the unsaturated moisture flow equation with application to evaporation from a water table. Soil Sci., 85(4): 228-232.

Green, W.H. and Ampt, G.A., 1911. Studies in soil physics: 1. The flow of air and water through soils. J. Agric. Sci., 4: 1-24.

Haverkamp, R., Parlange, J.-Y., Starr, J.L., ~chmitz, G. and Fuentes, C., 1990. Infiltration under ponded conditions: 3. A predictive equation based on physical parameters. Soil Sci., 149(5): 292-300.

Hillel, D. and Gardner, W.R., 1970. Transient infiltration into crust-topped profiles. Soil Sci., ! 09(2): 69-76.

Lantz, R.B., 1970. Rigorous calculation of miscible displacement using immiscible reservoir simulators. Soc. Petrol. Eng. J., 10(2): 192-202.

4 4 D.A. BARRY ET AL.

McWhorter, D.B., 1971. Infiltration affected by flow of air. Hydrol. Paper No. 49. Colorado State Univ., Fort Collins, CO.

Mein, R.G. and Larsen, C.L., 1973. Modeling infiltration during a steady rain. Water Resour. Res., 9(2): 384-394.

MiUy, P.C.D., 1985. Stability of the Green-Ampt profile in a delta function soil. Water Resour. Res., 21(3): 399-402.

Mualem, Y., 1976. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res., 12(3): 513-522.

Parlange, J.-Y., 1980. Water transport in soils. Ann. Rev. Fluid Mech., 12: 77-102. Parlange, J.-Y. and Hill, D.E. 1979. Air and water movement in porous media: compressibility

effects. Soil Sci., 127(5): 257-263. Reid, M.E. and Dreiss, S.J., 1990. Modeling the effects of unsaturated, stratified sediments on

groundwater recharge from intermittent streams. J. Hydrol., 114: 149-174. Richards, L.A., 1931. Capillary conduction of liquids through porous mediums. Physics, 1:

318-333. Rogers, C., Stallybrass, M.P. and Clements, D.L., 1983. On two phase filtration under gravity

and with boundary infiltration: Application of a Biicklund transformation. Nonlin. Anal. Theory Meth. Appl., 7(7): 785-799.

Sander, G.C., Parlange, J.-Y., Kfihnel, V., Hogarth, W.L., Lockington, D. and O'Kane, J.P.J., 1988. Exact nonlinear solution for constant flux infiltration. J. Hydrol., 97: 341-346.

Schweich, D., Sarding, M. and J.-P. Gaudet, J.-P, 1983. Measurement of a cation exchange isotherm from elution curves obtained in a soil column: Preliminary results. Soil Sci. Soc. Am. J., 47(1): 32-37.

Spanier, J. and OIdham, K.B., 1987. An Atlas of Functions. Hemisphere Publishing Corp., New York.

Sposito, G., 1989. The Chemistry of Soils. Oxford University Press, New York. Yortsos, Y.C., 1987. The relationship between immiscible and miscible displacement in porous

media. Am. Inst. Chem. Eng. J., 33(11): 1912-1915.

APPENDIX

The W function discussed in the text is useful for predictions of both infiltration and capillary rise. For practical use it is not convenient to solve the implicit function (16) to find W(x) for given x. The range of x, of interest here, is shown in Fig. 1 as - e x p ( - 1 ) < x ~< 0. The following present accurate approximations for W(x) in this range.

Case I: - 1 <<. W(x) ~< 0 (Capillary rise)

The relevant portion of the W function is labelled capillary rise in Fig. 1. Standard continued fraction methods yield the approximation:

W(x) ~ - 1 + r/ (A 1)

1 + N2+~/

0.10

• "-" 0 . 0 5

"~ 0.00

.>_

- 0 . 0 5

- 0 . 1 0 - 0 . 4 -0 .3 -0.2 -0.1 0.0

RICHARDS' EQUATION 45

Fig. AI . Relative error plot for the approximation given in the appendix for the upper portion of the W function in the range - e x p ( - 1) ~< x ~< 0.

where

r/ = x/2 + 2xexp (1 )

Ni = (1 X~2 )(N2 + x/'2) and

(A2)

(A3)

N 2 = 12.7036. (A4)

The relative error, defined as 100(W~x,¢t - Wapprox)/Wexact % , of this approxi- mation is plotted in Fig. A l, where the maximum relative error is less than 0.1% for all x in the range of interest. It should be noted that at x = 0, the approximation (A 1) is constructed so that the absolute error is 0 at x = 0, i.e. w ( o ) = o.

Infiltration

An approximation for the lower branch of W(x) is:

W(x) ,~ - 1

where

- - O"

2

M, ] 1 - M,x/- ~ (A5)

+ M2a exp(M3x/~ 1 +

1

a = - 1 - l n ( - x ) (A6)

46 D.A. BARRY ET AL.

0.03

ILl

O) .>_ o

n,-

0.02

0.01

0 .00

- 0 . 0 1

- 0 . 0 2

- 0 . 0 3 - 0 . 4

I I I

- 0 . 3 - 0 . 2 - 0 . 1 0.0

X

Fig. A2. Relative error plot for the approximation given in the appendix for the lower portion of the W function in the range - e x p ( - I) <~ x ~< 0.

M, = 0.3361 (A7)

M2 = -0 .0042 (A8)

and

M3 = -0.0201 (A8)

Figure A2 shows that the maximum relative error of(A5) is less than 0.025%. The maximum positive error of approximately 0.0252% occurs around x = - 2 x 10 -2~, and so is not apparent from Fig. A2. Thereafter, however, the relative error decreases. Equation (AS) is relatively simple to calculate. An even simpler formula with absolute relative error less than 0.35% is given by M, = 0.3205 and M2 = 0. In the latter case, the formula (with appropriate notation changes) is similar to the cumulative infiltration equation suggested by Brutsaert (1977, p. 367).