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WATER RESOURCES RESEARCH, VOL. 27, NO. 1, PAGES 109-117, JANUARY 1991 
Hillslope Infiltration' Planar Slopes 
J. R. ProLIP 
CSIRO Centre for Environmental Mechanics, Canberra, ACT, Australia 
The solution of the full nonlinear unsaturated flow equation is given for a problem of infiltration (and 
associated subsurface flows) into a planar hillslope of a homogeneous isotropic soil with uniform initial 
moisture content. The solution uses the assumption that, at some distance below the slope crest, a 
spatial equilibrium is approached with the moisture content (and other properties) independent of 
downslope coordinate x, and dependent only on the normal coordinate z, and time t. The analysis is 
found to be generally applicable beyond a small distance from the slope crest (or from a point of slope 
change). For slope angles less than 30 ø , infiltration normal to the slope differs relatively little from 
infiltration from a horizontal surface. Interesting aspects of the solution include a time-independent 
total horizontal flow into the slope and a time-dependent downslope total flow component which 
behaves as t 2/2 at small t and as t at large t. Previously, hillslope flows in these directions have been 
discussed in terms of soil anisotropy and layering, but these flows in a homogeneous isotropic soil are 
simple physical consequences of capillarity and gravity. The analysis applies with minor modification 
to two simple types of hillslope anisotropy: (1) anisotropy parallel to the slope and (2) horizontal 
anisotropy. Type 1 yields a complicated time dependence of the horizontal flow, and type 2 affects 
downslope flow similarly. These flow directions will commonly reverse during the course of the 
infiltration process. 
1. INTRODUCTION 
In recent decades, much attention has been paid to hill- 
slope hydrology, as attested by volumes edited by Kirkby 
[1978] and Abrahams [1986]. In this communication we 
develop extensions to infiltration theory for horizontal land 
surfaces [e.g., Philip, 1957b, c, 1969] needed to embrace 
hillslope conditions. Questions addressed include the follow- 
ing: 
1. How does surface slope affect the dynamics of infil- 
tration into the hillslope (i.e., normal to the slope)? 
2. What can be determined about the dynamics of the 
downslope (i.e., parallel to the slope) subsurface, essentially 
unsaturated, flow? The effects of hillslope divergence and 
convergence, and of concavity and convexity, are not ex- 
plored. Investigation of these aspects is in progress and will 
be communicated later. 
2. PROBLEM FORMULATION AND SOLUTION 
We consider a long planar hillslope of homogeneous 
isotropic soil. The slope angle is T. We introduce Cartesian 
rectangular space coordinates x and z, with x taken positive 
in the horizontal downslope direction and z positive in the 
downward vertical direction. We use also the rotated coor- 
dinates (x,, z,) defined by (Figure 1) 
x,=xcosy+zsiny (1) 
z, = -x sin T + z cos T 
The equation governing unsaturated soil water flow may 
be written [cf. Philip, 1957b, 1969] in the form 
oo dK oO 
--= V. (DV0) (2) 
Ot dO Oz 
Copyright 1991 by the American Geophysical Union. 
Paper number 90WR01704. 
0043-1397/91/90 WR-01704505.00 
Here t is time, 0 is the volumetric moisture content, D is the 
moisture diffusivity, and K is the hydraulic conductivity. In 
general, both D and K are strongly varying functions of 0. 
We investigate the dynamics of infiltration and downslope 
flow by seeking the solution of (2) subject to the conditions 
t=0 z,>0 0=0o 
(3) 
t>-0 z, =0 0 = 0• 
Here 00 is the uniform initial volumetric moisture content, 
and 0• is the value of 0 corresponding to moisture potential 
ß • at which water is available at the soil surface z, - 0. With 
free water present in excess at negligible depth on the 
hillslope, • = 0, and 0• is the saturated volumetric mois- 
ture content. 
Rewriting (2) in terms of x, and z,, we obtain 
--=V-(DV0) sin + ... cos •, (4) at - •' T OZ. 
Now, except in a small upper region of the slope (to be 
examined later), the relevant solution of (4) subject to (3) is 
essentially independent of x. and dependent only on z. and 
t. In these circumstances we may express (4) as 
--= D cos •/ (5) 
Ot Oz. dO Oz. 
We see that (5) subject to (3) is identical in form to the 
classical one-dimensional infiltration problem (e.g., (19) sub- 
ject to (20) of Philip [1957b]), except that K is replaced by K 
cos T- 
It follows that the series [Philip, 1957b] and traveling wave 
[Philip, !957c] solutions carry over to hillslope infiltration. 
The series solution, appropriate to small and moderate 
values of t, is 
z,(O,t) TM Z CPntn/2( cOS ,y)n-I 
rt= l 
(6) 
109 
110 PHILIP: HILLSLOPE INFILTRATION--PLANAR SLOPES 
z ..... •"" :•' "' '•"' '"•' ' ;'•....'•';• .-•.. 
Fig. 1. Schematic figure illustrating Cartesian space coordinates 
(x, z) and rotated coordinates (x., z.). Fig. 2. Schematic figure illustrating various flow velocity com- 
ponents: u, horizontal inslope; v, vertical; Ud, downslope; 
normal to slope. 
The "coefficients" •o•(0), •o•.(0), etc., are precisely the func- 
tions of 0 which enter the infiltration solution for a horizontal 
soil surface [cf. Philip, 1957a, b, 1969]. 
Similarly, the large t traveling wave solution is 
z.(O, t)= ut cos 'y + st(0) sec ¾ (7) 
Here 
Kl - K0 
u = ---- (8) 
0•- 00 
is the penetration velocity of the traveling wave, and st(0) the 
equation of the wave profile, for infiltration from a horizontal 
surface [cf. Philip, 1957c, 1969]. K•, K0, denote K(O•) and 
K(0o), respectively. We recall that 
g(0) = - 00) 
O, D dO ' (K! - K0)(0 - 00) - (K - K0)(0• - 00) (9) 
where the upper limit Oa (near, but not at 01) may be fixed by 
matching solutions (6) and (7) at relatively large t close to the 
limits of useful convergence of (6) [Philip, 1957c]. 
It is useful to restate the solutions in terms of the vertical 
space coordinate z, noting that it follows from the second of 
(1) that 
z = x tan 3,' + z, sec 3/ 
so we may express (6) and (7) in the alternative forms 
z(0, t) = x tan 3' + •'• qo ntn/2(cos 'y) n - 2 
n---1 
(lO) 
z(O, t) = x tan 3' + ut + •r(0) sec 2 y 
Note that with 0 = 0(z,, t), 
(11) 
00 00 00 00 
sin 7 .... cos 7 =- 
oz. o x oz. oz 
D = D sec 2 
Oz. •zz 7 
Equation (5) is thus equivalent to 
(12) 
m=_ D sec 2 7 
Ot Oz dO Oz 
(13) 
3. STANDARD AND ROTATED COMPONENTS 
OF UNSATURATED FLOW 
There are two separate and distinct means of resolving 
unsaturated flow associated with hillslope infiltration into 
two orthogonal components. See Figure 2. 
3.1. Vertical and Horizontal Components 
First, we may simply use conventional resolution of the 
flow velocity into horizontal and vertical components u and 
v, respectively. It is convenient here to take u positive 
directed into the hillslope. Then, in general, 
u = D 00/0 x > 0 (14) 
since, in general, O0/Ox > 0. That is, the horizontal flow 
velocity is directed into the hillslope. Also, the vertical 
component 
v = K- D O0/Oz (15) 
with v0, the value of v at the surface z = x sin 7, given by 
vo = K• - (D O0/OZ)z = xsin •, (16) 
Evidently, v0 is the vertical infiltration rate and is a volume 
flux per unit horizontal plan area. Defined thus, v0 is the 
infiltration rate of standard hydrologic practice. 
3.2. Normal and Downslope Components 
Students of hillslope hydrology, on the other hand, have 
an interest in the components of unsaturated flow parallel to, 
and normal to the slope, Ud and Vn. The use of n here should 
not be confused with the suffix n in series, for example, in (6) 
and (10). We take Ud positive in the downslope direction. 
Then 
Ud = --u cos 'y + v sin y (17) 
Vn = u sin y + v cos 3' (18) 
Using the first and second of (12), and combining (14), (15), 
(16), (17), and (18) then gives 
PHILIP: HILLSLOPE INFILTRATIONmPLANAR SLOPES 111 
0o 
u = -D • sin ? 
Oz, 
(19) 
O0 
v = K-D • cos ? 
•, 
[ vo=K1 - D 
Z, ----O 
(20) 
cos ? (21)ua = K sin ? (22) 
00 
Vn = K cos 3'- D • (23) 
Oz, 
We see that Vno, the value of v n at the surface z, = 0, is given 
by 
Vno=Klcos3'- D (24) 
Z,--0 
The quantity V no is the infiltration rate normal to the slope 
and is a volume flux per unit slope area. 
4. CUMULATIVE INFILTRATION AND INFILTRATION 
RATE FUNCTIONS 
4.1. Cumulative Infiltration and 
Infiltration Rate Normal to Slope 
With the infiltration solution expressed in the form of 
distance as a function of 0 and t, as by Philip [1957b, c, 1969] 
and in (6) and (7) of this paper, the cumulative infiltration and 
infiltration rate functions may be readily derived from the 
material balance. It follows here from (6) that the cumulative 
infiltration normal to the slope, 
in(t) = •', •ntn/2(cos 3')n-• + Kot cos 3' 
n--1 
(25) 
with 
•n = qo n dO (26) 
0 
Differentiation with respect to t gives the infiltration rate 
normal to the slope 
n nt(n/2) - l(cos T) n- 1 + K0 cos ? Vn0(t) = 
=1 
(27) 
The terms in K0 take account of the background flow at 
infinity in the case where Ko is not negligibly small [cf. 
Philip, 1957b, 1969]. 
We then have the truncated two-parameter expressions 
[cf. Philip, 1954, 1957d, 1987], useful for small and moderate 
values of t, 
in(t) = St 1/2 + At cos 7 (28) 
1 
Vno(t) = •St -v2 + A cos 3' (29) 
where the parameters S and A assume the values appropriate 
to infiltration from a horizontal surface. That is, as usual, S 
is the sorptivity [Philip, 1957d] defined by 
S = • (30) 
A = •2 + K0 (31) 
These various relations are supplemented by the traveling 
wave results, appropriate to large t, that 
lira in(t) = K•t cos 3' + Z sec 3/ (32) 
t--• o• 
lim Vno(t)= Ki cos 3' (33) 
The constant Z is the appropriate integral of • with respect to 
0. 
4.2. Vertical Cumulative Infiltration 
and Infiltration Rate 
It follows from (21) and (24) that 
vo(t) = Vno(t) cos T + K1 sin 2 ? 
(34) 
i(t) = in(t) cos ? + Kit sin 2 y 
Then application of (34) to (25), (27), (28), (29), (32), and (33) 
gives the various relations for i(t) and v0(t): 
i(t) = • CI>ntn/2(cos 3')n -b (K1 sin 2 3' + K0 ½OS 2 y)t 
n=l 
(35) 
vo(t) = ? •cI>nt(n/2)-l(cos3')n+Klsin2y+Kocos2y =1 
(36) 
i(t) = St 1/2 cos 7 + (A cos 2 3' + Kl sin 2 y)t (37) 
1 
vo(t) = •St -•/2 cos 3' + A cos 2 3' + K1 sin 2 3/ (38) 
lim i(t) = K•t + Z (39) 
lim v0(t) = K• (40) 
5. INTEGRATED HORIZONTAL AND DOWNSLOPE FLOWS 
5.1. Integrated Horizontal Flow 
The total horizontal flow per unit cross-slope length U has 
interesting properties. We have that 
U = udz = D•dz 
sin 3' sin 3' 0 X 
•o 00 •00x =- D dz, tan y = DdO tan 3' 
Oz, o 
(41) 
The third equality depends on the relations (Oz,/OX)o = 
-sin 3', (Oz,/OZ)o = cos ?. 
There are some notable aspects of this result. First, since 
112 PHILIP: HILLSLOPE INFILTRATIONmPLANAR SLOPES 
f oø• D dO 
is finite, U is finite even though the range of integration in z 
is formally infinite. Second, we may rewrite (41) as 
U = [O(01) - 6}(00)] tan )' (42) 
where O is the well-known Kirchhoff potential. Equation 
(41) thus reveals a new aspect of the physical significance of 
©. Third, and most remarkably, U is independent of t. That 
is, the rate of total horizontal discharge into the hillslope is 
constant throughout the infiltration process. 
This result is, at first glance, surprising, but it has a simple 
mathematical-physical explanation. As the horizontal flow 
velocity at fixed O, D 00/0 x, varies with t, the interval in z 
over which it is integrated to yield the contribution to U 
varies also with t and is proportional to Oz/00. Now O0/Ox is 
proportional to O0/Oz., but Oz/00 is proportional to its recip- 
rocal. The integral thus remains constant. Hence the contri- 
bution to U from each interval in 0 is independent of t, so U 
itself is independent of t. 
The constant Y may be fixed by matching series solution (45) 
at sufficiently large t. 
We observe that the inequality 
K-Ko O- 0o 
Ki - Ko 01 - 0o (48) 
which follows from the concavity of K(0) [cf. Philip, 1957c, 
1969, 1987], yields a useful inequality for Ua. Putting (48)in 
(43) gives 
ua(t) -oo V no(O) t--•oo in(O) 
Even for a slope angle as large as 30 ø, these ratios vary 
only between 1 at small t and 0.866 at large t. The slope 
produces a maximum 9ariation of about 13% and much 
smaller variations for small and moderate t values. See Table 
1. The slope effect is clearly minor. 
6.1.2. Moisture profiles. The slope effect on the mois- 
ture profiles is correspondingly small, but we remark on one 
aspect. At small t the moisture profiles for y = 0 and )' > 0 
are essentially identical. As t increases, moisture gradients 
at the wetting front become marginally less steep for )' > 0 
than for y = 0. In the limit of the traveling wave this effect 
attains its extreme. The whole wave shape then becomes 
less steep by the factor cos )' (= 0.866 for y = 30ø). 
PHILIP: HILLSLOPE INFILTRATION--PLANAR SLOPES 113 
TABLE 1. Effect of Slope Angle •/on Infiltration Rate vn0 and Cumulative Infiltration in, Computed for Yolo Light Clay 
, 
10 8 Vno, m s -I 10 3 i n, m 
Percent Percent 
t, s 103 •. ß = 0ø y = 30 ø Difference 7 = 0 ø • = 30 ø Difference 
5,000 4.8 93.48 92.85 0.67 9.10 9.07 0.33 
15,000 14.4 56.11 55.42 1.23 16.08 15.98 0.62 
50,000 48 33.18 32.44 2.23 30.53 30.18 1.14 
150,000 144 21.70 • 20.86 3.87 56.40 55.25 2.04 
500,000 480 15.13 14.08 6.94 117.18 112.74 3.94 
•o oo 12.30 10.65 13.40 13.40 
6.2. Inslope Horizontal Flow 
6.2.1. Horizontal flow velocity. It is of interest to ex- 
amine the dependence of the horizontal flow velocity u on 
both depth and time. Figure 3 shows the verticalprofiles of 
u for the Yolo light clay at t = 4 x 10 4 s and 5 x 10 5 s, with 
7 = 30ø- As t increases, u values tend to decrease but extend 
to greater depths. The total flow rate U (proportional to the 
shaded areas in Figure 3) is constant, independent of t. 
6.2.2. Total horizontal flow rate. For the Yolo light 
clay, (41) gives 
U = 3.404 x 10 -8 tan 7 m 2 s -1 (52) 
and its sorptivity 
S = 1.254 x 10-4 m s - 1/2 (53) 
108U (m s -1) 
0.2 
0.4 
0.6 
0.8 
1.0 
1.2 
1.4 
0 5 10 15 
5x105S 
0.48 
t = 4x104 s 
T = 0.0384 
Fig. 3. Vertical profiles of horizontal flow velocity u, computed 
for infiltration into Yolo lilght clay with slope angle y = 30 ø at t = 
4 x 104 s and 5 x 10 ø s, that is, at r = 0.0384 and 0.48. 
Dimensionless time r is defined in (58). The figure shows both the 
vertical space coordinate z - x tan y and also the dimensionless 
coordinate v, defined in (61). The shaded areas represent the 
horizontal flow rate U. They are equal since U is constant, indepen- 
dent of t. 
[Philip, 1957b]. It follows that (52) is equivalent to 
0.577 S 2 
U .... tan y (54) 
(0• - 00) 
As White and Sully [1987] pointed out, 
o• bS 2 DdO = • (55) 
Ol-0o 
o 
where the numerical constant b lies in the range 0.5 --- b - 6.2!9 x 10 5 
Figure 5 shows the Ua(t) function evaluated thus from (45) 
and (47). 
6.4. Generalizing the Results 
It has been convenient to develop illustrative numerical 
results for Yolo light clay with 7 = 30 ø. Obviously, the 
magnitude of slope effects varies with 7, and the appropriate 
1.2 
1.0 
0.8 
0.4 
0.2 
2 
i I ß 
•cy s t (s) 
i 
o.4 0.6 0.8 
lO 
Fig. 5. Dynamics of the integrated downslope flow rate Ua, 
calculated for Yolo light clay with slope angle 7 - 30ø- The figure 
shows both Ua(t) and the dependence on r of the dimensionless 
quantity Ua(01 - Oo)/(S 2 sin 7 cos 3'). 
equations in the foregoing express this in a direct and 
transparent manner. Furthermore, we stress that, although 
the numerical results are specific to Yo!o light clay, they 
indicate to a good approximation the mode of behavior of 
other soils. We may interpret the results more generally by 
expressing them in dimensionless form, so that they yield 
estimates relevant to any soil for which we know quantities 
such as (K• - K0), (0• - 00), and S, as well as 7. 
We thus make use of the characteristic infiltration time 
tgrav, introduced by Philip [1969] and defined by 
tgrav -- (57) 
K1 - K0 
Then the dimensionless time 
•' = titgray (58) 
Note that for the Yo!o light clay tgra v -- 1.04 x 106 s. 
The analogous characteristic infiltration length 
K1 - K0 S 2 
lgrav = tgrav = (59) 
Ol - 0o (K• - Ko)(O1 - Oo) 
In the different context of linearized multidimensional infil. 
tration, Philip [ 1986] used a characteristic infiltration length 
Xgrav, set equal to 2a -• , the sorptive length [Philip, !983]. 
White and Sully [1987] have shown that in the present 
symbolism 
2a -• = 2 b Igrav (60) 
Thus 2a -• and lgrav are of very similar magnitude, since 2b 
tends to lie in the range 1 to 1.2 [White and Sully, 1987]. We 
adopt the dimensionless space coordinates 
z- x tan 7 z, 
v = v. = (61) 
/gray /gray 
For the Yolo light clay, Igra v -- 0.497 m. 
Values of r as well as of t are given in Table 1 and on 
Figures 3-7; Figure 3 shows v as well as (z - x tan 7), and 
Figure 4 shows v. as well as z.. These facilitate application 
of the results to other soils. 
In studying Ua(t) it is useful to work with the dimension- 
less form 
Ua( O • - 0 o) Ua( O • - 0 o) 
(Kl - K0)2 sin y cos y tgra v S 2 sin 7 cos y 
(62) 
Note that it follows from (47) that 
d [ Ua(01-0o) ] lim -- . = 1 
•_•oodr S 2sinycos y 
(63) 
Figure 5, giving Ua(t), also shows the dimensionless Iorrn 
Ua(Ol - 0o)/[S 2 sin y cos y] as a function of r. 
Finally, we note that when U is expressed in the dimen- 
sionless form U(O• - 00) (cot y)/S 2, it reduces simply to the 
numerical constant b. This has the value 0.557 for the Yo10 
light clay and tends to lie in the range 0.5 to 0.6 for soils of 
interest in the present context. 
PHILIP: HILLSLOPE INFILTRATION--PLANAR SLOPES 115 
0.8 
0.6 
0.4 
0.2 
-0.2 
-0.4 
-0.6 
-0.8 
3O 
2O 
-1( 
-21 
-3O 
2 
4 
0 2 4 6 8 
1(• 5 t (s) 
0 0.2 0.4 0.6 0.8 
T 
lO 
Fig. 6. Type 1 anisotropy: Dynamics of integrated horizontal 
flow rate U, computed for infiltration into soil with K(0) and D(0) as 
for Yolo light clay in the normal direction but with/•K(0) and #D(0) 
parallel to the slope, with slope angle 3/= 30 ø. The curves show 
results for anisotropy/• = 0.5, 1, 2, and 4. The figure shows both 
U(t) and the dependence on r of the dimensionless quantity U(01 - 
00) (cot 7)/S 2. Compare section 6.3. 
7. PHYSICAL DISCUSSION 
1.0 
0.5 
2O 
-0.5 
-1.0 
-1.5 
0 2 4 6 8 !0 
1(•5t (S) l 
0 0.2 0.4 0.6 0.8 1.0 
Fig. 7. Type 2 anisotropy: Dynamics of integrated downslope 
flow rate Ua, computed for infiltration into soil with K(0) and D(0) as 
for the Yolo light clay in the vertical direction but taK(O) and/.09(0) 
in the horizontal, with slope angle 7 = 30 ø. The curves show results 
for anisotropy/• = 0.5, 1, 2, and 4. The figure shows both Ua(t) and 
the dependence on r of the dimensionless quantity Ud(01 - 0o)/(S 2 
sin 7 cos y). 
7.1. Applicability of the Analysis 
Central to our relatively simple analysis of hillslope infil- 
tration is the assumption that, beyond some point x. = X. 
far enough down the slope from the crest x. = 0, spatial 
equilibrium is attained with both the infiltration normal to the 
slope, and the downslope subsurface flow, independent of x. 
and with 0 = 0(z., t). 
The material balance for 0 -• x, -• x.] requires that the 
total net rate of inflow of water into the wetted region, less 
the time rate of increase of the total water in the region, 
equal the integrated downslope flow Ud(x.•). That is, 
X, 1 Ud(X*l) : [VnO(X.) - Ko cos 7] dx, 
d foX, 1 - • fo• Odz,dx, (64) 
For spatial equilibrium, Ud(x, 1) = constant = Ua, indepen- 
dent of x,]. This requires that U,i be a relatively small 
constant difference between the two near-equal quantities 
represented by the integrals in (64), each of which increases 
in magnitude approximately as x, •. Accordingly, the equi- 
librium requires that 
fo X* UdKo)(in(t) - Kot cos y) 
X. >> sin 7 (66) 
(01 -- O0)(VnO(t) -- Ko cos y) 
But, from (28) and (29), 
i•(t) - Kot cos y 
lim ...... t•o vno(t) - Ko cos • 2t (67) 
and, from (32) and (33), 
in(t) - Ko t cos y 
lim , 
t•o Vno(t) - Ko cos ? 
= t (68) 
It follows that the right side of (65) varies from 2(K• - Ko)t 
(sin 7)/(0] - 00) at small t to (K• - Ko)t (sin y)/(0• - 00) at 
large t. It is thus conservative to use the inequality 
X, >> 2(K1 - Ko)t (sin y)/(01 -- 00) (69) 
116 PHILIP: HILLSLOPE INFILTRATION--PLANAR SLOPES 
We adopt, as the practical criterion for applicability of the 
analysis, 
X. > 40(K1 - Ko)t (sin 3/)/(0• - 00) (70) 
For our numerical example this gives 
X, > 9.54 x 10 -6t m (71) 
with t in seconds. We see that only the top 0.95 m of the 
slope is excluded for t = 105 s and only 9.54 m for t = 106 s. 
We note the general result that for t = tgra,,, (70) becomes 
just 
X, >- 40/gray sin 3/ (72) 
These criteria indicate that our analysis is generally appli- 
cable beyond a small distance from a slope crest (or below a 
point of slope change). This small distance is roughly pro- 
portional to t and may typically reach about 10 m at an 
advanced stage of the infiltration event. Note that/gray tends 
to lie in the range 0.1 to 2 m [cf. White and Sully, 1987]. 
7.2. Resolving the Flow into Components 
We have treated at some length the distinction between, 
on the one hand, horizontal and vertical flow velocity 
components u and v and components parallel and normal to 
the slope, ud and Vn. Stressing this distinction and evaluating 
the various components is not a vacuous pedantic exercise. 
Indeed, the value of certain hillslope hydrology studies has 
been decreased by confusion on this matter. For example, 
Harr [1977] discussed hillslope seepage in terms of the 
vertical flow velocity (our v) and downslope flow velocity 
(our u d). He calculated the resultant flow through a false 
analogy with the parallelogram of forces. His work, which 
centered on the magnitude and direction of the resultant 
flow, is thus vitiated. Various subsequent writers on hills- 
lope infiltration cite this study uncritically. 
7.3. Physical Features of Hillslope Infiltration 
Some simple general statements can be made about the 
physical basis of the differences between hillslope infiltration 
and infiltration from a horizontal surface. These involve 
reference to the physical basis of infiltration dynami. cs, 
namely, the interaction between capillary and gravitational 
effects [Philip, 1969]. 
For infiltration normal to the slope, capillary effects are 
essentially the same as for a horizontal surface, since X70 is 
normal to the slope, but the component of gravity normal to 
the slope is reduced by the factor cos y, and gravitational 
effects are reduced accordingly. On the other hand, for 
vertical infiltration from a sloping surface it is the capillary 
effects which are reduced by the factor cos % while gravi- 
tational effects remain essentially as for the horizontal sur- 
face. These differences find expression in the different forms 
of equations such as (28) and (29) for normal infiltration and 
(37) and (38) for vertical infiltration and in the difference of 
each of these from equations for the horizontal surface 
(given by putting y = 0). 
7.4. Inslope Horizontal Flow 
A superficially remarkable result of the analysis is the 
uncovering of the horizontal flow component into the hill- 
TABLE 3. Limiting Power Dependence on t of Various 
Hillslope Infiltration Quantities 
Power of t in Power of t in 
Quantity Small t Limit Large t Limit 
vno - 1/2 0 
in 1/2 1 
U 0 0 
Ud 1/2 1 
slope. Further interesting aspects are that the integrated flow 
rate U is constant, independent of t, and is proportional to 
ølD dO (compare (41)). These results are, however, ele- J'00 
mentary consequences of the fact that, on the hillslope, 
capillary effects have a component directed horizontally into 
the slope. 
So far as I can ascertain, previous hillslope studies [e.g., 
Zaslavsky and Rogowski, 1969; Zaslavsky and Sinai, 1981; 
McCord and Stephens, 1987] have discussed horizontal flow 
components in terms of anisotropy or soil layering. A point 
of interest here is that our analysis is for a homogeneous 
isotropic soil. We need invoke neither anisotropy nor layer- 
ing to produce in-slope horizontal flow. In section 8 we 
address briefly modifications to the present result conse- 
quent on anisotropy. 
7.5. Downslope Flow 
We have seen that, for a homogeneous isotropic soil, there 
is a time-dependent downslope flow Ue proportional to t m 
in the small t limit and to t in the large t limit. This flow is the 
elementary physical consequence of the fact that there is a 
downslope component of gravity. 
Downslope flow has been described previously [e.g., 
Zaslavsky and Rogowski, 1969; Zaslavsky and Sinai, 1981; 
McCord and Stephens, 1987] in the context of anisotropy or 
soil layering; but here also neither is needed. Gravity suf- 
fices. We see in section 8, however, that anisotropy may 
complicate the dynamics of U•(t) considerably. 
7.6. Overall Dynamics 
The foregoing analysis yields, inter alia, the limiting power 
dependence on t of various flows associated with hillslope 
infiltration, for both small and large t. This offers a useful 
overall picture of the dynamics of the various processes and 
is presented in Table 3. 
8. EFFECTS OF ANISOTROPY 
With minor modifications, the foregoing analysis applies 
also to two types of anisotropic hillslope. For both types the 
anisotropy/x is assumed to be constant, independent of the 
moisture potential. In type 1, anisotropy parallel to the 
hillslope, we take K(O) as the component of the conductivity 
tensor normal to the hillslope and/xK(0) as the component 
parallel to the slope. In type 2, horizontal anisotropy, we 
adopt K(0) as the vertical component of the conductivity and 
ttK(O) as the horizontal component. For each type the 
anisotropy of D is the same as for K. 
PHILIP: HILLSLOPE INFILTRATION---PLANAR SLOPES 117 
8.1. Type 1: Anisotropy Parallel to the Slope 
In this case, (22) is replaced by 
ua =/xK sin ? (73) 
but (23) is unchanged. It follows at once that 
ua(/x) =/xua(1) Ua(tx) =/xUa(1) (74) 
where ua(/x), Ua(/x) signify values for anisotropy/x. Criterion 
(70) for applicability of the analysis becomes 
X, >- 40/x(Kl - Ko)t (sin 7)/(01 - 00) (75) 
The effect of anisotropy on downslope flow is simply to 
magnify it by the factor/x. Equations (5) and (13) remain 
unchanged as the partial differential flow equations in terms 
of z, and z, respectively, and solutions such as (6) and (7) 
still apply. The results for Vno and in are unaltered. 
On the other hand, type 1 anisotropy changes the charac- 
ter of the dynamics of horizontal flow. Combining (17), (18), 
(23), and (73), we find 
O0 
u =-D sin ,/- (/x - 1)K sin y cos ¾ (76) 
Oz, 
Whereas for/x = 1, u > 0, we find that for/z > 1, u changes 
sign as both t and z vary. Details are easily inferred from 
(76), but we concentrate here on the dynamics of U(t). We 
note that to make our definition of U consistent with that of 
Ua (in subtracting out background seepage associated with 
initial hillslope moisture content) we require here that 
U = [u + (/z - 1)K0 sin y cos y] dz (77) 
sin •, 
Then (76) gives 
U(/x) = U(1) - (/x - 1)Ua(1) (78) 
where U(/x)denotes U for anisotropy tz. Whereas U was 
constant and independent of t for /x = 1, it becomes 
time-dependent for/x • 1. Figure 6 compares the dynamics 
of U(t) for/x = 0.5, 1, 2, and 4. 
8.2. Type 2' Horizontal Anisotropy 
In this case (19) is replaced by 
00 
u = - It/) sin y (79) 
but (20) is unchanged. Then 
u(/x) =/au(1) U(/x) =/xU(1) (80) 
where u(/x) denotes u for anisotropy •. The effect of type 2 
anisotropy on horizontal flow is simply to magnify it by the 
factor •. U remains a constant, independent of t. 
On the other hand, putting (78) and (20) into (17) gives 
00 
u• = Ksin ,/+ (/a - 1)D • sin ,/cos ,/ (81) 
Oz. 
Whereas for Ix = 1, ue is inherently positive, for/a > I it may 
change sign as both t and z vary. Details may be found from 
(81), but it suffices here to examine the dynamics of Ua(t). It 
follows from (81) that 
Ua(/x) Ua(1) (tt 1) U(1) cos 2 = - - y (82) 
Figure 7 compares the dynamics of Ua(t) for/a = 0.5, 1, 2, 4. 
Finally we note that type 2 anisotropy changes the partial 
differential flow equations (5) and (13) to 
•=• D (•sin 2y+ cos 2•) 
at Oz, dO Oz, 
cos 'y 
(83) 
- D (/.• tan 2 y + 1) dO Oz (84) ot oz 
Our established methods of solving equations of this form 
apply. 
Acknowledgments. I am grateful to my colleague I. White for 
helpful discussions. I also thank the Australian Water Research 
Advisory Council for the Eminent Researcher Fellowship which 
supported this work. 
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J. R. Philip, Centre for Environmental Mechanics, CSIRO, GPO 
Box 821, Canberra, ACT 2601, Australia. 
(Received January 29, 1990; 
revised June 5, 1990' 
accepted June 20, 1990.)