Soil Science Society of America Journal 64:24-29 (2000)
© 2000 Soil Science Society of America
DIVISION S-1-SOIL PHYSICS
A New Solution for Water Storage to a Fixed Depth for Constant Flux Infiltration
Bing Cheng Sia and
R.Gary Kachanoskia
a Dep. of Soil Science, Univ. of Saskatchewan, Saskatoon, SK, Canada S7H 5A8
gary.kachanoski{at}usask.ca
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ABSTRACT
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A quantitative description of water infiltration under constant flux boundary conditions is useful for predicting water flow and chemical transport processes occurring in surface soils. A new quasi-analytical solution for water storage to a fixed depth for constant flux infiltration is presented. The solution, based on the flux–concentration relationship, allows general forms of soil hydraulic functions to be used and provides a direct interpretation of measurements from vertically installed time domain reflectometry probes. To evaluate the solution, a constant flux infiltration experiment was conducted and the hydraulic parameters for the experimental site were determined independently. The solution predictions using the approximate flux–concentration relationship for linear soils were essentially identical to an existing analytical solution using the Broadbridge and White form of hydraulic functions and to the measurements.
Abbreviations: TDR, time domain reflectometry
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INTRODUCTION
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A QUANTATIVE DESCRIPTION OF WATER INFILTRATION under constant flux boundary conditions in unsaturated soils is fundamental to understanding water balance, irrigation, movement of chemicals and, more generally, transport processes occurring in surface soils. Despite the success of numerical solutions, analytical solutions have received considerable attention because they are very useful for assessing the accuracy of numerical models and provide insight into the physics of flow phenomena. Additionally, analytical solutions can be used to test inverse techniques for non-uniqueness and identifiability of hydraulic parameters of interest. In the past 30 years, analytical solutions of Richards' equation for constant flux water infiltration into homogeneous soil profiles have been developed using approximate integral procedures (Parlange, 1972; Philip and Knight, 1974; White et al., 1979) and exact transform methods such as Kirchhoff, Hopf-Cole and Storm transformations (Broadbridge and White, 1988; Warrick et al., 1990), and reciprocal Bäcklund transform by Sander et al. (1988, 1990).
The nonlinear Richards' equation was solved by Parlange (1972), who described an approximate integral procedure for the solution of infiltration by exploiting the rapid change of diffusivity with water content. Philip and Knight (1974) showed how Parlange's method could be improved to any desired accuracy through the use of a concept called the flux–concentration relation (Philip, 1973). The use of the flux–concentration relation, in principle, permits quasi-analytical solution of the highly nonlinear flow equation to be found for a wide range of flow phenomena in soils. White at al. (1979) analyzed constant-flux adsorption using an approximate flux–concentration relation. Experiments using a fine sand validated the approach and indicated that both the surface water content and the water content profile could be predicted accurately for the horizontal adsorption of water supplied to the sand at a wide range of constant flux rates. Perroux et al. (1981) extended the solution to constant-flux infiltration and concluded that sufficiently accurate predictions of soil water profile development can be made by using the simple adsorption analysis of White et al. (1979). Boulier et al. (1984) confirmed the ability and the versatility of the flux–concentration relation-based approach to predict water infiltration into soils.
Exact solutions for infiltration were developed for linear soils (Braester, 1973). Such a linearized solution can only be expected to approximate the integral properties of the soil–water system. Parlange (1976) pointed out a significant disparity between surface water contents calculated from this linearized solution and those calculated numerically. In addition, the linear convection term does not permit the development of a traveling wave solution at large infiltration times. This problem does not arise in the exactly solvable Burgers' equation with its weakly nonlinear convection term. The solution to Burgers' equation satisfactorily described rainfall infiltration in an undisturbed field soil (Clothier et al., 1981). However, like the linear soil, Burgers' solution treats diffusivity as constant, even though soil water diffusivity varies over several orders of magnitude across the water content range of interest. Broadbridge and White (1988) and Sander et al. (1988) independently presented exact analytical solutions for constant-flux infiltration based on realistic nonlinear dependence of unsaturated hydraulic conductivity and Fujita-type diffusivity on soil water content. These solutions not only accurately predict water infiltration but also produce all the salient features of water flow during constant-flux infiltration, including the traveling wave solution.
Parkin et al. (1992) derived an analytical solution for water storage to a fixed depth based on the analytical solution of Broadbridge and White (1988) and Sander et al. (1988). The model result can be used directly to interpret the measurement of water storage from vertically installed TDR (time domain reflectometry) probes. However, the solutions of Parkin et al. (1992) require specific forms of diffusivity and hydraulic conductivity dependence of water content in the Broadbridge and White (1988) solution. This limits its applications where hydraulic parameters are known only for other forms of hydraulic properties such as the van Genuchten and Mualem (van Genuchten, 1980), Brooks and Corey (Brooks and Corey, 1966), and Gardener and Russo (Russo, 1988) forms. The objective of this paper is to present a quasi-analytical solution for water storage to a fixed depth during constant-flux infiltration, based on the solution of White et al. (1979). This solution allows functions for general soil hydraulic properties. We compare our solution with the solution of Parkin et al. (1992) and with the measurements from a field experiment.
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Theory
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We consider nonhysteretic vertical soil water flow under constant water application at the soil surface and seek to find an expression for time dependence of water storage to a fixed depth. The flow of water is described in this process by the continuity equation,
 | (1) |
and Darcy's Law,
 | (2) |
where, t (s) is time, z (m) is the vertical coordinate, 
(m3 m-3) is the volume water content, q (m s-1) is the volumetric flux of water, D() (m2 s-1) is the water content-dependent soil water diffusivity, and K() (m s-1) is the hydraulic conductivity.
Substituting Eq. [2] into Eq. [1] yields the nonlinear Richards' equation used to describe one-dimensional nonhysteretic flow in idea soil:
 | (3) |
 | (4) |
 | (5) |
The initial and boundary conditions considered here are the uniform initial water content, n, the constant surface water application rate R (m s-1) on the soil surface with Ks (m s-1), the saturated hydraulic conductivity.
An Analytical Solution Based on White et al. (1979)
Philip (1973) introduced the flux–concentration relation, F(
0,t) as
 | (6) |
where
, 0 is the surface water content. And Kn (m s-1) is the initial soil hydraulic conductivity. Substituting Eq. [6] into Eq. [2] and integration with respect to gives (White et al., 1979):
 | (7) |
We use the identity
 | (8) |
to transform Eq. [1] into
 | (9) |
Integrating with respect to , leads to (White et al., 1979)
 | (10) |
Substituting z(,t) from Eq. [7] into Eq. [10] and integrating by parts yields
 | (11) |
Water storage at time t, W(L,t) to a fixed depth
, may be obtained by integrating z(,t) by parts with respect to :
 | (12) |
where L(t) is the water content at depth
as a function of time, t. Substituting z(,t) from Eq. [7] into Eq. [12] and integrating by parts leads to
 | (13) |
Equations [7], [11], and [13], together give a quasi-analytical solution for constant-flux infiltration. If F(,t) is known, Eq. [7] and [11] can be used to predict the time dependence of the surface soil water content 0(t) and
at depth
, respectively. With 0(t) and L(t) known, Eq. [13] can be used to predict the change of water storage with time. All these calculations can be done easily with Mathcad (Mathsoft, Cambridge, MA). This quasi-analytical solution for water storage to a fixed depth L is general in terms of the form of D() and K(). It allows us to use the more versatile van Genuchten water retention curve (Eq. [14]) combined with the Burdine hydraulic conductivity function (Eq. [15]) (van Genuchten, 1980):
 | (14) |
with
, where n and 
VG are fitting parameters, respectively. Ks is the saturated hydraulic conductivity.
 | (15) |
Here, we selected the Burdine hydraulic conductivity function because it fits our data better than the Mualem hydraulic conductivity. However, our solution applies to both functions.
This solution is for constant-flux infiltration into soil of uniform initial water content; however, it can also apply to constant-flux infiltration into a soil of nonuniform initial water content with modification of F(0,t), 0(t), and L(t).
Generally, we need the iterative procedure of Philip and Knight (1974) to obtain F(0,t). However, it is well known that the time dependence of F(0,t) is negligible (White et al., 1979; Philip, 1973). The extreme cases of soil hydraulic properties are those of constant diffusivity (Linear soil and Burgers' soil) and a Dirac function (Green and Ampt soil). For linear soil, F(0) is exact and can be approximated by
. For Green and Ampt soil, F(0) is also exact and equal to 0 for constant concentration adsorption. Philip (1973) conjectured that for constant-flux infiltration, F(0) lies in the narrow band bounded by
.
Solution of Parkin et al. (1992)
Broadbridge and White (1988) and Sander et al. (1988) independently developed analytical solutions for constant flux infiltration. The Broadbridge and White solution is based on the following parameterization of hydraulic conductivity and diffusivity functions:
 | (16) |
 | (17) |
where
and
. s and r are the saturated water content and residual water content, respectively. Ks, , and C are the saturated hydraulic conductivity, inverse capillary length scale (Philip, 1985), and a constant introduced by Broadbridge and White. From Eq. [16] and [17], Broadbridge and White derived the water retention function as
 | (18) |
where 
0 is an integration constant. Following Broadbridge and White, we set
.
Using Eq. [16] and [17] through a series of transforms (i.e., Kirchhoff, Storm, and Hopf and Cole transforms), Broadbridge and White solved Eq. [3], [4], and [5] and derived an analytical solution as
 | (19) |
and
 | (20) |
where 
is a parameter connecting Eq. [19] and [20], and u(,t) is given in Warrick et al. (1990) as follows:
 | (21) |
 | (22) |
 | (23) |
By change of variable of integration, Parkin et al. (1992) obtained an analytical solution for water storage to depth L for constant flux infiltration,
 | (24) |
To calculate W(L,t), the first step is to evaluate through Eq. [20] by setting
. The second step is to calculate u(,t) using Eq. [21]. Then W(L,t) can be calculated through Eq. [24].
The Broadbridge and White model encompasses a wide range of realistic soil hydraulic properties by varying the C parameter. As C goes to infinity, the model reduces to the weakly nonlinear Burgers' equation, which has been applied in certain field conditions. At the other end of the range as C approaches 1, the Broadbridge and White model approaches the Green-Ampt-like model (White and Broadbridge, 1988).
In the following, we take advantage of the dimensionless variables for the Broadbridge and White (1988) form of the hydraulic function
 | (25) |
This transforms Eq. [13] and [24] into equations involving only the C parameter (Parkin et al., 1995). Therefore, the sensitivity of F(0) to different soils can be examined through changes of the C value.
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Materials and methods
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Field infiltration measurements were conducted at the Canadian Forces Base Borden, Ontario, Canada. Extensive hydrogeological research, including a large-scale, natural-gradient tracer test and forced-gradient test, have been conducted by University of Waterloo on this site (Sudicky, 1986). The infiltration experiments have been discussed in detail by Si et al. (1999). Briefly, water was applied to an instrumented transect (7.5 m long) inside a greenhouse using a hanging track and nozzle system. Multipurpose TDR probes were installed every 0.15 m at each of four depths (0.2, 0.4, 0.6, and 0.8 m) for a total of 200 TDR probes. Five different water application rates were used. Soil water content was measured using the TDR method of Topp et al. (1980). The readings were taken manually from the display screen of two precalibrated Tektronix 1502 C (Tektronix, Wilsonville, OR) metallic cable testers by four operators. The readings were taken just before the start of water application and every 5 to 30 min, depending on infiltration rate and rate of change of , for all the 200 multipurpose TDR probes. Here, we use the site average of the 50 probes for the 0.2-m depth as an illustration. A fit of van Genuchten and Broadbridge and White models for () and K() to measured data is given in Si et al. (1999). The parameters for the Broadbridge and White model are
,
,
, and
. The parameters for the van Genuchten model are
,
,
,
, and
.
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Results and discussion
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Figure 1
(a–f) depicts dimensionless water storage–time functions for the new solution and the Parkin et al. (1992) solution for the limiting conditions;
, appropriate for a repacked coarse material (Green and Ampt soils); and
, appropriate for soils with a wide range of pore sizes. The initial increase in storage is clearly linear for all soils, reflecting the constant surface-applied infiltration rate. As the wetting front moves below L, the rate of change of storage gradually decreases for the Burgers' soil and abruptly reaches equilibrium for the Green and Ampt soils.
For
, 1.02, and 1.10, the new solutions with
and
are essentially identical to each other and to the solution of Parkin et al. (1992) when the Broadbridge and White forms of K() and D() are used. This suggests that the F(0), for either a linear soil or the Green and Ampt soil, is accurate enough to predict water storage during constant flux infiltration. It also indicates the new solution is not sensitive to the value of F(0) at small C. This is reasonable, since diffusivity changes abruptly with water content as the value of C gets close to 1. Thus, the selection of F(0) from the narrow band is not critical. However, when C increases to 1.5 as in Fig. 1d, diffusivity changes gradually with water content and the gravity effect becomes significant. Thus, the dominance of D() over F(0) is lost and the influence of different F(0) in the new solution becomes significant. This becomes obvious at
(Fig. 1e) and
(Fig. 1f), where different F(0) results in significant differences in the predicted water storage in the middle curve region. For these C values, the new solution with
for linear soil is essentially identical to the solution of Parkin et al. (1992), while the new solution with
significantly underestimate the water storage relative to the solution of Parkin et al. (1992). As expected, the predictions for all solutions at initial time and large time are identical for different C values, since the increase of water storage with time is a reflection of applied flux density, while the prediction at large time reflects the water content at
. Thus, the main difference in the models are reflected in the curvature of W(L,t).
White (1979) found that the time dependence of F(0,t) for constant flux adsorption into Bangendore fine sand was negligible and that the measured F(0,t) lies only slightly above the line
. Boulier et al. (1984) pointed out that the measured flux concentration relation can be well approximated by
and the time dependence is not significant. These experiments were based on repacked coarse materials and it is reasonable to infer that the C values for the materials would be close to 1. Thus, it is not difficult to understand why the predictions using both
and
were successful for the prediction of surface water content and the water content profile during constant-flux adsorption and vertical infiltration (White et al., 1979; Perroux et al., 1981; Boulier et al., 1984). However, based on the above analysis, we suggest that
be used since it applies to most field soils and gives identical results to the Parkin et al. (1992) solution, while
only applies to repacked laboratory and coarse field soils.
Application to Field Data
For
, the new F(0) solution (Eq. [7], [11], and [13]) using either Broadbridge and White or van Genuchten parameters as given earlier, gives almost identical predictions of water storage vs. time to the measurements and to the Parkin et al. (1992) (Eq. [24]) (Fig. 2)
. The predictions from all models are highly correlated with the measurements (Table 1)
. The underestimation using the van Genuchten model is more than that using the Broadbridge and White model, suggesting that different forms of hydraulic models have different sensitivity to the form of F(0). The major difference in the solutions, as expected, are in the middle curved regions, which reflect the integrated effect of the wetting front shape. An alternative choice of F(0) could be made by using
, where 0 < ß < 1 (Kutilek, 1980). An optimal ß could be calculated by matching the Broadbridge and White solution and the new solution presented here. The average difference between predicted and measured average water content (water storage divided by the length of TDR rods) is <0.01 for all solutions, which is smaller than the measurement error of TDR (Topp et al., 1980).
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Table 1 Statistics for measured vs. the predicted water storage to depth
using the solution of Parkin et al. (1992), the new solution with, Broadbridge and White (BW) model, and the new solution with van Genuchten (VG) model for
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Summary and conclusion
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A new quasi-analytical solution for water storage to a fixed depth was presented. The solution is based on the flux–concentration relationship of Philip (1973). Using the approximate F(0) relationship of White et al. (1979) for linear soil, the solution is essentially identical to that of Parkin et al. (1992), however, for a wide range of realistic hydraulic properties. The solution was applied to a field soil. Using independently measured soil hydraulic properties, the predictions were essentially identical to the measurements.
Received for publication January 26, 1999.
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[Abstract]
[Full Text]
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TOP
ABSTRACT
INTRODUCTION
, (c)
, (d)
, (e)
