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WATER RESOURCES RESEARCH, VOL. 27, NO. 6, PAGES 1035-1040, JUNE 1991 
Hillslope Infiltration' Divergent and Convergent Slopes 
J. R. PHILIP 
CSIRO Centre for Environmental Mechanics, Canberra, Australia 
This perturbation analysis of divergence/convergence effects on hillslope infiltration and downslope 
unsaturated flow indicates that, in general, these effects are unimportant for usual hillslope topogra- 
phy. Only where the radius of the contour is less than about 10 times the characteristic infiltration 
length/gray (or 10 times the sorptive length) need the perturbation be taken into account. This result 
opens up a considerable extension of applicability of the analysis of infiltration on planar hillslopes to 
embrace slopes with curved contours, such as ridge noses and hollows. 
1. INTRODUCTION 
Hillslope infiltration and downslope subsurface unsatur- 
ated flow for planar slopes has been studied recently [Philip, 
1991]. Here we address the question of how results of that 
work are affected by slope divergence or convergence. 
Figure 1 indicates schematically what we mean by these 
terms. Figure 1 a shows a planar slope, with equally spaced 
linear contours; Figure 1 b shows a divergent slope, with the 
radius of curvature of the concentric contours increasing 
downslope; and Figure l c shows a convergent slope, with 
the radius of curvature of the concentric contours decreasing 
downslope. Both the divergent and convergent slopes are (at 
least part of) conical surfaces with the cone axis vertical. 
!ida [1984] studied saturated flow in such conical slopes. He 
referred to divergent slopes as "ridge-type" and to conver- 
gent ones as "hollow-type". Previously, Beven [1977], using 
an essentially two-dimensional model, parametrized diver- 
gence/convergence through a "slope width". It will be 
understood that our divergent slopes are relevant to the 
slopes of isolated hills and to the horizontally convex slopes 
at ridge noses, and our convergent slopes model those of 
isolated closed depressions and the horizontally concave 
hollows at valley heads. 
2. PROBLEM FORMULATION: 
SLOPE DIVERGENCE PERTURBATION 
We consider a long divergent hillslope of homogeneous 
isotropic soil with uniform slope angle 3'. We use cylindrical 
coordinates (r, z) with the origin of the horizontal radial 
coordinate r on the axis of the conical surface which 
contains the hillslope. Note that r is the radius of curvature 
of the slope contours. The vertical coordinate z is taken 
positive downward. In the radial half-plane (r, z) we also use 
the rotated coordinates (r,, z.) defined by 
r.=r cos •/+z sin 3• 
(1) 
z, = -r sin •/+ z cos y 
(see Figure 2a). 
The relevant axisymmetric form of the equation governing 
unsaturated soil water flow [cf. Philip, 1969, equation (25)] is 
Published in 1991 by the American Geophysical Union. 
Paper number 91WR00098. 
1035 
•=-- D + D -• , (2) at Or •zz r Or dO Oz 
where t is time, 0 is volumetric moisture content, D is 
moisture diffusivity, and K is hydraulic conductivity. In 
general, D and K are strongly varying functions of 0. We 
seek the solution of (2) subject to the conditions 
t=0 z.>0 0=00 
(3) 
t>0 Z.=0 0=01 . 
Here 00 is the uniform initial moisture content, and 0• is the 
0 value corresponding to moisture potential • at which 
water is available at the soil surface z, = 0. With free water 
present in excess at negligible depth on the hillslope, •1 = 0, 
and 0• is the saturated moisture content. 
The corresponding equation for the planar hillslope 
[Philip, 1991, equation (2)], with x replaced by r, may be 
written as 
(4) 
We see that (4) is identical to (2) except for the term 
Dr-•OO/Or on the right of (2). This term represents the 
primary effect of slope divergence. For all t this slope 
divergence term is at most of order r-• relative to (at least 
the largest of) the other terms on the fight of (2). Divergence 
thus gives a perturbation of the solution for planar slopes of 
order r -• . Evidently, the appropriate perturbation solution 
will be usefully convergent for r large enough, that is, far 
enough from the cone axis. 
Philip [1991] simplified the analysis for long planar slopes 
by seeking the solution which depends only on t and the 
rotated coordinate z,. Equation (4) was in this way reduced 
to the simpler [Philip, 1991, equation (5)] 
• = cos y. (5) 
The solution of (5) subject to (3) was 
Z,(O, t) m E qontn/2( cOS ,y)n-l, 
n=l 
(6) 
1036 PHILIP: HILLSLOPE INFILTRATION' DIVERGENT AND CONVERGENT SLOPES 
a Planar slope 
Cone 
axis 
b Divergent slope 
!. 
ß .:.;.:,::. -.::;:.:.::'-" 
c Convergent slope 
Fig. 1. Schematic figures showing (a) a planar slope, •b) a 
divergent slope, and (c) a convergent slope. The arrows indicate the 
direction of maximum slope. In Figures lb and lc the vertical axis 
of the conical surface is shown. In both cases the contours are 
centered on this axis. 
with the coefficients •p,(`0) precisely the functions of 0 
entering the series solution for infiltration into a horizontal 
soil surface [Philip, 1957a, b; 1969]. 
In the present work we develop the perturbation analysis 
by injecting into (5) the divergence term Dr -] O0/Or in an 
appropriate approximate form. With (6) taken as the leading 
term of the required solution, we find that, to order r -I , 
D 00 D 00 
_ m = sin y. (7) 
r Or r o Oz, 
Here r 0 is the value of r at the surface point (r,, z,) = (r0 
sec y, 0). Using the right side of (7), we thus obtain the 
perturbed form of (5)' 
.... D - sin y+•cos y az, (8) •t Oz, ' 
This equation, accurate to order r• • , enables us to evaluate 
the leading term of the divergence perturbation. 
We therefore seek the solution of (8) subject to (3) of the 
form 
z,(0, r 0, t)=f0(0, t)+ r•f•(O, t). (9) 
The expansion may be taken to higher powers of rg l, but 
this is not required for present pu•oses. It is convenient to 
write f0 for the z,(0, t) of (6), which is z,(0, m, t) in the 
extended notation of (9). 
Putting (9) into (8) and equating coefficients of rg • gives 
the required equation in f], namely, 
•=• D +D sin (10) 
3t 80 30] Y' 
This is subject to the conditions 
a Divergent slope 
Cone 
axis 
Cone 
axis b Convergent slope 
Fig. 2. Schematic figures illustrating space coordinates (r, z) 
and rotated coordinates (`r,, z,)' {a) divergent slopes, and {b) 
convergent slopes. 
o, f , dO = D 0-•o 0 o 0=0 ! 
dt 
+ D dO t sin y 
0 
t>-0 fl=0 0= 0•. (11) 
The first condition in (11) preserves the material balance to 
order r• -]. The second follows from the identity of the 
second condition in (3) (this paper) with the second condition 
in (3) of Philip [1991]. Integrating (10) with respect to both 0 
and t, and using (11), we obtain the numerically convenient 
integrodifferential form 
fl dO- D •dt + D dO t sin y, 
O0 
o o 
(,12) 
PHILIP: HILLSLOPE INFILTRATION: DIVERGENT AND CONVERGENT SLOPES 1037 
subject to the condition 
t :> 0 f• = 0 0 = 01. (13) 
3. DIVERGENT SLOPES: SERIES SOLUTION 
Putting (6) into (12) and (13), we see by equating coeffi- 
cients of powers of t 1/2 that f• is of the form 
f •(O, t) = Z •n(O )tn/2(cos 7) n- 2 sin T, 
n=2 
(14) 
CONVERGENT SLOPES 
The analyses for convergent hillslopes go similarly. In this 
case we consider a long convergent hillslope of homoge- 
neous isotropic soil with uniform slope angie y (see Figure 
2b). We take cylindrical space coordinates (r, z) as before, 
and again we use the rotated coordinates (r., z.) in the 
radial half-plane. Here, however, the sense of rotation is 
reversed, and we therefore replace 7 in equation (1) by -7- 
In what follows we shall not usually distinguish between 
the divergent and convergent cases; it will be understood 
that 7 > 0 for slope divergence and 7t • in (12) gives the equation for se2: 
/o • d • 2 f o • + DdO •2 dO = P dO ' 
o o 
(15) 
with P the function of 0 defined by 
P = D(dO/dqo •) 2. (16) 
Equation (15) is subject to the condition 
0 = 0• •2 = 0. (17) 
Similarly, equating coefficients of t 3/2 gives the equation for 
•3. It is 
• •:3 dO = P d•3/dO - Q, 
o 
(18) 
with Q the function of 0 defined by 
00 0½2 0•2 
Q = 2D .... . (19) 
0q0 1 Oc.P 10qO 1 
Equation (18) is subject to the condition 
0 = 0• •3 = 0. (20) 
Repeating this procedure gives the integrodifferential 
equations for •4, •5, etc. The method of solution and the 
form of the resulting equations closely follow those of the 
infiltration series solutions [Philip, 1957a, b; 1969]. The 
simple and rapid numerical technique for solving the se- 
quence of linear integrodifferential equations [Philip, 1957a] 
applies here also. Note that (15) is similar to equation (10) of 
Philip [1966] for m = 2. It tums out that (in present notation) 
-•2(0) was evaluated there in the context of radial two- 
dimensional absorption. 
Finally, we combine (6), (9), and (14) to secure the 
required solution to order r• -• , namely, 
) •2 tan 7 z,(O, r0, t) = qOl tl/2 + qøn + •n 
_ r0 
ß (COS 7) n- ltnl2. (21) 
5. FURTHER CALCULATIONS 
5.1. Flow Velocity Components and 
Integrated Flows 
Philip [1991] used two different means of resolving soil 
water flow velocity: First, resolution into horizontal and 
vertical components, u and v, with u positive directed into 
the hillslope and second, resolution into components parallel 
and normal to the slope, u d and V n, with u d positive 
downslope. We employ both sets of components here also. 
See section 3 and Figure 2 of Philip [1991]. We note, in 
particular, that in the present symbols 
u = D 0 O/Or, (22) 
ud = K sin 7, (23) 
and, in the infiltration rate norma/to the slope, 
v,•0 = K• cos 7- (D O0/Oz.)z.=o. (24) 
Philip [1991] also introduced the integrated horizontal flow 
U, defined in the present symbols by 
U = u dz, (25) 
sin y 
and the integrated downslope flow U a' 
Ud = (u d--Ko sin 7) dz. = (K-Ko) dz. sin 7 
(26) 
We study these integrated flows here also. 
5.2. Normal Infiltration 
The computations rapidly grow complicated, but it suffices 
to treat only the leading term of the perturbation, calculable 
from •2 for the series solution. We denote the result for 
planar slopes by Vno(O•, t) (i.e., r o -, oo) [Philip, 1991]. We 
find that the series solution yields 
Vno(ro, t)- Vno(O•, t)- D dO- •2 , (27) 
ro 
o 
where 
dO. 
1038 PHILIP: HILLSLOPE INFILTRATION: DIVERGENT AND CONVERGENT SLOPES 
Note that asymptotic large t analysis is more complicated 
here than for planar slopes [Philip, 1991] and is not pursued. 
However, 
lim Vno(ro, t)= Kt cos y, (28) 
so that 
lim [vno(r o, t) - Vno(O•, t)] = 0. (29) 
Divergence decreases v•0 and convergence increases it, 
though, as we shall see, the effects are often trivial in the 
applications. 
5.3. Horizontal Flow 
K1 -- Ko K• - Ko 
a = = ß (34) 
q'• K d• D dO 
o o 
•0 is the moisture potential corresponding to moisture 
content 00. For a given soil situation the two lengths are 
connected by the relation 
2a-l= 2b Igrav, , (35) 
where, as shown by White and Sully [1987], the constant b 
ranges from 0.5 to 0.6 under many conditions of practical 
interest. We develop the argument here in terms of l•av. 
The magnitude of the perturbations may be inferred from 
the magnitude of the leading term of the series expansion of 
f•, and specifically from the magnitudes of •2, E2, and 
together with that of f 0ø0 • D d 0. 
It may be readily shown that, to order r• 'l, the total 
horizontal flow 
6.1. Magnitude Estimates and Computed Values 
for the Example 
• O0 •= O0 U= D • dz = - D • dz tan ¾ 
Or sin • sin 7 
= D dO tan 7- 
o 
(3O) 
To this order, the total horizontal flow U is unaffected by 
divergence or convergence. It is constant and independent of 
t, as for planar slopes [Philip, 1991]. 
5.4. Integrated Downslope Flow 
Here also we limit ourselves to the leading term of the 
perturbation. U d(OO, t) denotes the result for planar slopes 
[Philip, 199!]. Then the series solution gives 
Ud(ro, t) -- Ud(o:, t) = r•-lA 2 t sin 2 y, (31) 
with 
The integrodifferential equation for •2, (!5), and the con- 
dition governing it, (equation (17)), are identical to those for 
qo2 [Philip, 1957a, equations (3.17) and (3.18); Philip, 
1957b, equations (33) and (34); Philip, 1969, equations (88) 
and (92)] except that the term K - K0 is replaced by f 00 D 
dO. Since (15) is linear, it follows that 
q>2(0-----• = 0 D dO/(K 1 - Ko) (36) 
0 
[Io •2= 0• 
cb: 0 D dO/(K• - Ko) . (37) 
• 0 
0• We recall that •: = f Oo •o: d 0. Now it follows from (34) and 
(35) that 
S' D dO/(K• - Ko) = b /•rav, o 
so (36) and (37) become 
(38) 
O= 0o A 2 = (K(O) - Ko) ds•2 . (32) 
dO=Or 
Note that divergence increases the integrated downslope 
flow, though it decreases the normal infiltration rate. Con- 
vergence gives reverse effects. 
6. THE MAGNITUDE OF THE PERTURBATIONS 
On a priori physical grounds one expects the perturbation 
due to divergence/convergence to be trivially small provided 
r is large compared with characteristic lengths of the flow 
process. One such length is/•rav, defined [Philip, 1991] by 
S 2 
/gray = (K1 ' ro)(Ol - 00)' (33) 
Another is the sorptive length 2a -• , where the sorptive 
number a is defined [Philip, !983] by 
•2( 0 ) 
qo2(0'-• = O[b /srav], (39) 
• = O[b Igrav]- (40) 
(I) 2 
We may check these relations for the illustrative example 
of Philip [1991] for Yo!o light clay with 00 = 0.2376, 0• = 
0.4950. For that soil, b = 0.557 and l•rav = 0.497 m, so (40) 
becomes •2/(I) 2 = 0[0.277 m]. Calculation gives ,E2 = 
!.3808 x 10 -8 m 2 s -• . Now, since (I)2 = 4.654 x 10 -8 m s -• 
[Philip, 1957b], we find •2/(I) 2 = 0.297 m, agreeing well 
with (40). 
Satisfaction of (40) implies satisfaction of (39), so long as 
the •:2 (0) and q•2 (0) functions are reasonably similar in shape. 
This condition is tested in Figure 3, which compares se2(0) 
and b/•rav ½2(0) for the Yolo light clay example. 
We need also the magnitude of A2/t{ 2, where K2 is the 
second coefficient in the series expansion for Uo(oo, t) 
[Philip, 199!]. Comparing the relation 
PHILIP: HILLSLOPE iNFILTRATION' DIVERGENT AND CONVERGENT SLOPES 1039 
01 0 o 
o 
i - 
2 - 
4 
I I I I I I ...... I ,,, I I ,, I 
Fig. 3. Comparison of the divergence/convergence perturbation 
function •2(0) (solid curve) with bigray •2(0) (dashed broken curve), 
both computed for Yolo light clay. The order of magnitude agree- 
ment is consistent with (39). 
"O= 0o 12(A - K0 in the symbolism 
of that paper) is given for Ko/Ki small by 
•2 = c(K1 - Ko). (43) 
The constant c ranges from 0.3 to 0.4 for common soil 
situations. Combining (38) and (43) then gives 
f o•' D dO/•2 = b lgrav/c. o 
Then (44) and (40) yield 
(44) 
• D dO - •2 o 
(I) 2 
= O[b(c-i _ 1)/gray], (45) 
an order statement requiring that (40) be sufficiently accu- 
rate. The cited ranges of b and c imply that the factor 
b(c -1 - 1) lies in the range 0.75 to 1.4. For the Yolo light 
clay example, c = 0.378, so (45) gives (f 
= 0[0.456 m]. We have, further, that the calculations for 
Yolo light clay [Philip, 1991] give f •o • D dO = 3.404 x 10 -8 
ß ot D dO- •2)/(I)2 is thus m 2 s -1 The actual value of (œOo • 
0.435 m, agreeing well with (45). 
6.2. Normal InfiltrationRate 
We see from (27) that the primary effect of the perturba- 
tion is to change the second term of the series expansion for 
Vn0- The perturbation is established for small and moderate 
values of t, but (29) indicates that it decreases to zero as t ---> 
o•. It is apparently of greatest significance at moderate values 
of t. 
The perturbation is evidently negligible for all t when r0 is 
sufficiently large. We take as our criterion for this that the 
perturbation should not exceed 10% of the t o term of the 
unperturbed expansion for V no [Philip, 1991, equation (27)]. 
This requires that 
D dO-E2 tan 7 
0 
r0 > 10 . (46) 
•2 
Combining this with (45) gives 
r0 > 100[b(c -1- 1)/•rav] tan 3'- (47) 
Adopting the upper bound on b(c -1 - 1), 1.4, and remov- 
ing the order sign gives the suggested criterion 
r o > 14 Igrav tan 7, (48) 
which reduces, for 3' = 30 ø, to 
r 0 > 8.08 lgrav- (49) 
Detailed calculations for the Yolo light clay example with 3' 
= 30 ø give 
r 0 > 5.05 lgrav = 2.51 m, (50) 
consistent with (48). 
6.3. Integrated Downslope Flow 
Here also the primary effect of the perturbation is to 
change the second term of the series expansion for U a. It is 
evidently negligible for all t when r 0 is large enough. We 
adopt the criterion that the perturbation should not exceed 
10% of the t 1 term of the unperturbed expansion for Ua 
[Philip, 1991, equation (45)]. This requires that 
3,2 
r0 -> 10 • tan 3'. (51) 
I 10 O[b /grav] tan 3'- (52) 
Adopting the upper bound on b, 0.6, and removing the order 
sign gives 
r 0 -> 6/gray tan 3', (53) 
a criterion less stringent than (48). The detailed calculation 
for Yo!o light clay gives 
1040 PHILIP: HILLSLOPE INFILTRATION: DIVERGENT AND CONVERGENT SLOPES 
r 0 -> 5.5 /grav tan y = 2.75 tan •, m, 
in agreement with (53). 
(54) 
6.4. Discussion 
The foregoing results give the r0 ranges for which diver- 
gence/convergence perturbations may be neglected. We see 
that, provided r0 exceeds these various bounds, the analysis 
for planar slopes [Philip, 1991] is applicable. Since/gray tends 
to lie in the range 0.1 to 2 m [cf. White and Sully, 1987], the 
bounds we have established imply that typically the pertur- 
bation is negligible so long as r0, the radius of the contour, 
exceeds a few meters. 
These bounds, however, must be taken in conjunction 
with the condition for assured accuracy of the planar analy- 
sis on which this perturbation study is based. This [Philip, 
1991 ] is that 
X. > 40(K• - Ko)t sin ?/(0• - 00), (55) 
with X. the downslope distance from the slope crest (or a 
point of slope change). This quantity translates into 40(K• - 
Ko)t cos ), sin ?/(0• - 00) as a horizontal interval in r. It 
tends to be of the same order of magnitude as the bounds 
examined above. 
For divergent slopes the bounds (48) and (55) both apply at 
the upper end of the slope, with the larger overriding, and 
there is no bound at the lower end. For convergent slopes, 
on the other hand, (55) fixes the bound at the upper end, and 
(48) does so at the lower end. 
7. CONCLUDING DISCUSSION 
7.1. The Character of the Perturbations 
Although the perturbations are small, it is useful to sum- 
marize their signs and their physical basis. For divergent 
slopes the depth of moisture penetration is increased 
slightly. This is accompanied by a slight reduction in the 
surface moisture gradient and hence also in the infiltration 
rate Vno. On the other hand, the slightly increased normal 
depth of penetration gives a small increase in the cross 
section for downslope flow and therefore also in the total 
downslope flow Ud. The signs of these various modifications 
are reversed for convergent slopes. 
7.2. Extended Applicability of Planar Analysis 
This perturbation analysis of the effects of hillslope diver- 
gence/convergence has indicated that, in general, they are 
unimportant for usual hillslope topography. Only where the 
radius of the contour is less than about 10 times the charac- 
teristic infiltration length lgrav (or 10 times the sorptive 
length) need the perturbation be taken into account. This 
result opens up a considerable extension of applicability of 
the analysis of infiltration on planar hillslopes [Philip, 1991] 
beyond purely planar slopes to slopes with curved contours 
(e.g., on ridge noses and in hollows). 
7.3. Physical Remark 
In physical terms the explanation of this result is clear. It 
is that divergence/convergence adjustments to the flow oc- 
cur on the scale of hillslope topography (the contour radius 
r), whereas the unsaturated flow processes driven by capil- 
larity and gravity can respond on the generally much smaller 
scale of/gray (or the sorptive length). In consequence, the 
divergence/convergence perturbations to the planar slope 
results tend to be negligibly small. 
Acknowledgtnents. I am grateful to Ian White for helpful discus- 
sions and to Cathy Wilson for drawing my attention to Beven (1977) 
and Iida (1984). I also thank the Australian Water Research Advi- 
sory Council for the Eminent Researcher Fellowship which sup- 
ported this work. 
REFERENCES 
Beven, K., Hillslope hydrographs by the finite element method, 
Earth Surf. Processes, 2, 13-27, 1977. 
Iida, T., A hydrological method of estimation of the topographic 
effect on the saturated throughflow, Trans. Jpn. Geomorphol. 
Union, 5, 1-12, 1984. 
Philip, J. R., Numerical solution of equations of the diffusion type 
with diffusivity concentration-dependent, II, Aust. J. Phys., I0, 
29-42, 1957a. 
Philip, J. R., The theory of infiltration, 1, The infiltration equation 
and its solutions, Soil Sci., 83,345-357, 1957b. 
Philip, J. R., Absorption and infiltration in two- and three- 
dimensional systems, in Proceedings, lASH/UNESCO Sympo- 
sium, vol. 1, Water in the Unsaturated Zone, edited by R. E. 
Rijtema and H. Wassink, pp. 503-525, UNESCO, Paris, 1966. 
Philip, J. R., Theory of infiltration, Adv. Hydrosci., 5, 215-296, 
1969. 
Philip, J. R., Infiltration in one, two, and three dimensions, in 
Advances in Infiltration, Proceedings of the National Conference 
on Advances in Infiltration, Chicago, U.S.A., pp. 1-13, American 
Society of Agricultural Engineers, St. Joseph, Mich., 1983. 
Philip, J. R., Inverse solution for one-dimensional infiltration and 
the ratio A/K•, Water Resour. Res., 26, 2023-2027, 1990. 
Philip, J. R., Hillslope infiltration: Planar slopes, Water Resour. 
Res., 27, 109-117, 1991. 
White, I., and M. J. Sully, Macroscopic and microscopic capillary 
length and time scales from field infiltration, Water Resour. Res., 
23, 1514-1522, 1987. 
J. R. Philip, Centre for Environmental Mechanics, CSIRO, GPO 
Box 821, CANBERRA ACT 2601, Australia. 
(Received March 9, !990; 
revised October 3, 1990; 
accepted January 11, 1991.)