Infiltration

Infiltration is the process by which liquid water enters soil. Through redistribution, water that entered the soil during infiltration is redistributed within the soil. Both infiltration and redistribution profoundly affect the soil water balance. If water were applied to the surface of a uniform column of air-dry soil and the rate of infiltration of water into the soil column were measured as a function of time, the infiltration rate would be shown to decrease with time. These features are depicted in Fig. 8.2. The infiltration rate is high initially, but decreases with time to a constant value. If a similar experiment were done with a column of moist soil, similar results would be obtained, but the initial rate would be lower. If infiltration were into a horizontal, rather than a vertical, column, once again similar results would be obtained, except that the infiltration rate would decrease towards zero, rather than the constant, non-zero value for a vertical column like the one depicted in Fig. 8.2. If water content were measured at several times during infiltration, the water content profiles in the column would be similar. Two features of these profiles

Infiltration 167

0.0 –0.8 –1.0

1 h

3 h 5 h –0.6

–0.4 –0.2 0.0

Depth [m]

0.10 0.05 0.00 0.15 0.25 0.20 0.30 0.35

0.1 0.2 0.3 0.4 0.5

Water content (m³ m^–3)

Infiltration rate [kg m–2 s–1]

0 (a)

(b)

1000

500 1500 2000 2500 3000 3500

Time [s]

Figure 8.2 (a) Water content profiles at three times during infiltration.

(b) Infiltration rate as a function of time.

should be noted. First, there is a zone of almost constant water content extending from the soil surface to the apparent boundary between wet and dry soil. This zone is called the transmission zone. A visible wetting front or boundary between wet and dry soil exists at the lower end of the transmission zone. This sharp front is the result of the sharp decrease in hydraulic conductivity with water content that is characteristic of unsaturated porous materials.

A quantitative analysis of infiltration is obtained by solving the differential equation that describes water potential in soil as a function of position and time. The observa- tions described above are qualitatively consistent with predictions one might make from Darcy’s law:

vertical flow: f_w= –Kdψm

dz +Kg (8.8)

horizontal flow: f_w= –Kdψm

dx (8.9)

Equation (8.8) is for vertical flow, with the second term on the right-hand side be- ing the gravitational component of the water potential gradient. Equation (8.9) is for

168 Transient Water Flow

horizontal flow. As water first infiltrates a vertical column,dψm/dzis large, so the flux is large. As the length of the transmission zone increases, the absolute value ofdψm/dzde- creases. When the transmission zone is very long, the matric potential gradient becomes negligible, so the flux becomesf_w=Kg. Since the transmission zone is near saturation,K is nearK_s. For horizontal infiltration,f_wapproaches zero as the matric potential gradient approaches zero. The fact that infiltration rate is initially lower in moist soil is explained by the reduced matric potential gradient in moist soil. These qualitative descriptions can be formalized in a more quantitative way that allows for quantification of infiltration rates, through an infiltration model.

Philip’s model

The model most frequently referred to is by Philip (1957). In a one-dimensional soil profile, the infiltration ratev_I is defined as

v_I =dI

dt (8.10)

whereI is the cumulative infiltration [kg m^–2]:

I = t

0

v_Idt (8.11)

The cumulative infiltration corresponds to the increment in soil water content in the soil volume. For a simple case of infiltration into a horizontal column, it is written as

I = _θup

θlow

x dθ (8.12)

where the integration is performed between two limits of water content, here denoted by θlowandθup. For specific initial and boundary conditions, Philip analytically solved theθ- based form of Richard’s equation (8.7). He transformed the partial differential equation into an ordinary differential equation by assuming a relationship between space and time, where x =χ√

t, allowing for substitution of the variablesxandtwithχ.

Using this transformation of variables, the variable x in the integral (8.12) can be written as

I = _θup

θlow

χt^1/2dθ=t^1/2 _θup

θlow

χdθ =S_I√

t (8.13)

where S_I is the value of the integral, called sorptivity. Applying this definition to the infiltration velocity, which is the derivative ofI with respect to time, we obtain

v_I = d(S_I√ t) dt = S_I

2√

t (8.14)

Infiltration 169 so the infiltration velocity decreases inversely with the square root of time. The constant of proportionality is the sorptivity

S_I = _θup

θlow

χdθ (8.15)

This solution is valid for infiltration into a horizontal column. Infiltration into a vertical column implies that the integral of the total water content entering the soil surface must be equal to the changes in soil water content in the soil column below plus the water leaving the soil column at the bottom (mass balance). These differences are due to the fact that the water potential in a partially saturated water column is at a value of matric potential, but the gravitation potential decreases towards the bottom boundary with a water potential gradient equal tog.

For infiltration into a vertical column, the definition of cumulative infiltration is I = –

_θup θlow

z dθ+K_θlowt (8.16)

The negative sign in front of the integral is due to the fact that as depth increases, water content decreases, causing the integral to become negative. Now it is possible to perform the variable substitution proposed by Philip (1957):

I =t^1/2–χdθ+t

K_θlowt+ _θup

θlow

(–χ)dθ

+t^3/2 _θup

θlow

(–χ)dθ

+· · · (8.17) where the integral of the different elements of the series depends on the initial soil condition and soil type. They can therefore be written as

I =At^1/2+Bt+Ct^3/2+· · ·+Mt^m/2 (8.18) where A is equal to the sorptivity for a horizontal column, while the other terms are constant for a specific problem and soil. For relatively short times, the series in eqn (8.18) converges rapidly. Philip (1957) proposed that for the majority of cases, the two terms are sufficient to provide a robust solution

I =S_It^1/2+Bt (8.19)

and

v_I = S_I

2t^1/2 +B (8.20)

The infiltration process, as described by this model, is strongly affected by the value of S_I for small values of t, while for large values of t, the series in eqn (8.18) does not converge. On the other hand, a variety of experiments confirm the proportionality of the process to√

t.

170 Transient Water Flow

8.3.1 Green–Ampt Model

A simpler approach was taken much earlier by Green and Ampt (1911). Their solution gives the most important features of the infiltration process, so it will be examined in detail.

During horizontal infiltration, an observable wetting front or boundary moves through the soil. The water content (and therefore the water potential) at the wetting front is almost constant during infiltration, as long as the profile is uniform. If the dis- tance from the soil–water boundary to the wetting front isx_f and the potentials at the boundary and wetting front areψiandψf, respectively, then the infiltration rate (or flux at the boundary) is

f_wi= –Kdψm

dx = –Kψf–ψi

x_f (8.21)

whereKis the mean conductivity of the transmission zone.

The amount of water stored in the soil per unit time is the change in water content of the soil from its initial dry condition to the average water content of the transmission zone, multiplied by the rate of advance of the wetting front. To satisfy continuity, all of the water flowing in must be stored by advance of the wetting front, so

Kψi–ψf

x_f =ρlθdx_f

dt (8.22)

Hereρl is the density of water,θ= (θi+θf)/2 –θo,θi,θf andθo are the volumetric wa- ter contents at the inflow, wetting front and unwetted soil, andtis time. Separation of variables and integration gives the position of the wetting front as a function of time:

x_f =

2K(ψi–ψf)t

ρlθ (8.23)

Equation (8.23) indicates that the distance to the wetting front is directly proportional to the square root of time. The infiltration rate is obtained by combining eqns (8.21) and (8.23) to obtain

f_wi=

ρlθK(ψi–ψf)

2t (8.24)

showing that the infiltration rate is directly proportional toθ^1/2andK^1/2and inversely proportional tot^1/2. Equation (8.24) can be integrated over time to find the cumulative infiltration. The result of this integration is

I =

2ρlθK t(ψi–ψf) (8.25)

Numerical Simulation of Infiltration 171 The Green–Ampt equation can also be applied to vertical infiltration, but the result is somewhat less satisfying. The most important result is that, for long times,f_wiapproaches a constant value.

8.3.2 Infiltration into Layered Soils

The analysis so far has been concerned only for infiltration into uniform soils. Such soils exist mainly in carefully prepared laboratory columns. In the field, processes of soil de- velopment, tillage, illuviation, etc. produce layering. When infiltrating water encounters a boundary between layers, the infiltration rate generally decreases. The infiltration rate decreases for either sand or clay layers. The decrease with a clay layer is expected, be- cause water infiltrates clay less readily than coarser-textured materials, so water slows down as it enters the clay, decreasingψi–ψf and thus decreasing infiltration rate. The saturated hydraulic conductivity of sand is higher than that of clay, while the unsaturated hydraulic conductivity of sand is lower than the unsaturated hydraulic conductivity of finer-textured materials. The boundary must therefore be wet almost to saturation before the hydraulic conductivity of the sand increases sufficiently to start carrying the water away. This increase in water content at the boundary decreasesψi–ψf and decreases f_wi. The infiltration rate recovers after water enters the sand, but does not recover for the clay. Perched water tables are therefore common over clay layers, but are not possible over coarse-textured materials. In this chapter (and in the numerical solution), we have included the option of setting up the simulation for a soil having horizons with different properties. The user can select any number of soil horizons.