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DIVISION S-1-SOIL PHYSICS

Unified Solution for Infiltration and Drainage with Hysteresis

Theory and Field Test

^a Soil Science Dep., Univ. of Saskatchewan, Saskatoon, SK, Canada S7N 5A8

gary.kachanoski{at}usask.ca

	ABSTRACT

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Conclusion REFERENCES

 
Hysteresis has been found in both the hydraulic conductivity, K, vs. pressure head, , relationship, and the soil water content, , vs. relationship. This limits the application of a unified solution for infiltration and drainage. A Haines' Jump model of hysteresis is proposed and combined with the Broadbridge and White form of K() and the diffusivity, D, relationship, D(). This allows a unified analytical solution for infiltration and drainage. This solution accounts for hysteresis by allowing the inverse macroscopic capillary length scale, , to be hysteretic. A method of a priori estimating the hysteretic nature of is proposed and tested. The hysteretic change in can be estimated from other () hysteresis models and then used in combination with the Broadbridge and White hydraulic functions. The predicted hysteresis in was similar to that obtained from inverse procedures. The unified solution was applied to field-measured soil water storage during infiltration and drainage. Neglecting hysteresis resulted in poor prediction of water storage during drainage based on hydraulic parameters estimated from infiltration. This was especially true for drainage with high initial water content. Incorporating the proposed hysteresis model resulted in prediction error less than measurement error. In addition, a single unified inverse procedure for estimating hydraulic parameters from combined infiltration and drainage measurements can now be developed.

Abbreviations: RMS, root mean square error • TDR, time domain reflectometry

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Conclusion REFERENCES

 
A QUANTITATIVE 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. Analytical solutions of Richards' equation for constant-flux water infiltration into homogeneous soil profiles have been developed using integral procedures (Parlange, 1972; Philip and Knight, 1974; White et al., 1979), Kirchhoff, Hopf-Cole and Storm transforms (Broadbridge and White, 1988; Warrick et al., 1990, 1991), and reciprocal Bäcklund transform (Sander et al., 1988, 1991; Barry and Sander, 1991). Parkin et al. (1992, 1995) presented analytical solutions for water storage to a fixed depth based on solutions of Broadbridge and White (1988) and Warrick et al. (1990). These analytical solutions are useful for assessing the accuracy of numerical models and to estimate soil hydraulic properties by inverse procedures (Si et al., 1999). Analytical solutions can also be used to test various inverse techniques for uniqueness and identifiability of various hydraulic parameters of interest. The model results of Parkin et al. (1992, 1995) can be used directly to interpret time domain reflectometry (TDR) measurements.

To quantitatively predict the movement of water through variably saturated soils, detailed knowledge of the hydraulic properties of the soil are needed. The unsaturated hydraulic conductivity, K, expressed as a function of the soil water content, , or the soil water pressure head, , and the relation between and must be specified before analytical or numerical models can accurately predict water flow during infiltration, evaporation, or drainage. Unfortunately, because of hysteresis, these relationships are not simple functions and show a great deal of variation between wetting and drying cycles. Hysteresis has been found in both () and K() (Haines, 1930; Staple, 1969; Kool and Parker, 1987; Jaynes, 1992). Studies suggest that there is little hysteresis in K() or it is so slight as to be masked by the error of the measurements and can be ignored (Gillham et al., 1976; Topp, 1971).

Considerable effort has been put into the analysis and description of hysteretic soil hydraulic properties. This has led to numerous models for describing hysteresis in () (Gillham et al., 1976; Scott et al., 1983; Mualem, 1974, 1984; Kool and Parker, 1987; Parlange, 1976; Hogarth et al., 1988). These models provide a simple means for determining scanning curves from a limited amount of data, such as the main wetting and drying hysteresis curves. The models of Parlange (1976) and Mualem (1984) need only one branch of the loop to predict all scanning curves. Viaene et al. (1994) compared different models of hysteresis using 10 measured scanning curves and concluded that the best models were the conceptual models needing two branches for calibration. Simulation studies carried out by Jaynes (1985, 1992) have shown that none of the models were consistently better than the others. Numerical simulations of flow during transient infiltration and redistribution using a variety of hysteresis models did not differ greatly and agreed reasonably well with experimental water distribution, even when the scanning curves were not described very accurately (Kool and Parker, 1987).

Unfortunately, all the models, empirical or theoretical, do not allow exact unified analytical solutions of infiltration and drainage, even though the hydraulic models of Broadbridge and White (1988) and Sander et al. (1988) allow exact solution independently for infiltration and drainage. As a result, completely different sets of parameters have to be used for infiltration and drainage. This greatly inhibits the use of the analytical solution and our understanding of the role of hysteresis in the application of infiltration and drainage. In this paper, we present a model of hysteresis that connects the analytical solution of infiltration with that of drainage, thus allowing a unified solution of both drainage and infiltration. We apply the model to field-measured water storage during infiltration and drainage. To test the approach, the hydraulic parameters estimated from infiltration are used to predicted measured soil water storage during drainage.

	Theory

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Conclusion REFERENCES

 
Broadbridge and White (1988) and Sander et al. (1988) independently developed an analytical solution for constant flux infiltration. The Broadbridge and White solution is based on the following parameterization of hydraulic conductivity, K(), and diffusivity, D(), functions:

(1)

(2)

where and . _s is the saturated water content and _r is the residual water content. K_s, , and C are, respectively, the saturated hydraulic conductivity, the inverse capillary length scale (Philip, 1985), and a constant introduced by Broadbridge and White. By definition,

(3)

Thus,

(4)

Substituting Eq. [1] and [2] into Eq. [4] and integration yields

(5)

where ₀ is an integration constant. Following Broadbridge and White, we set .

The nonlinear Richards' equation can be used to describe one-dimensional non-hysteretic flow in ideal, homogeneous, isotropic, rigid soils:

(6)

where, t is time, z is the vertical space coordinate, is the volume water content, and D() is the water content–dependent soil water diffusivity.

The initial and boundary conditions considered here are

(7)

(8)

where .

Using Eq. [1] and [2] through a series of transforms (i.e., Kirchhoff, Storm, and Hopf and Cole transforms), Broadbridge and White, as well as Warrick et al. (1990), derived an analytical solution as

(9)

and

(10)

where is a parameter connecting Eq. [9] and [10], u(,t) is given by Eq. [43] of Broadbridge and White, and

(11)

By changing the variable of integration, Parkin et al. (1992, 1995) obtained an analytical solution for water storage to depth L for constant-flux infiltration and drainage:

(12)

where is hysteretic; (L,t) and u(,t), as functions of , are also hysteretic. In addition, this equation only applies to flow under uniform initial conditions.

Hysteresis Models
Haines' Jump Hysteresis Model
Hysteresis is caused by a change of energy status of water when a wetting process is switched to a drying process or vice versa. The change in energy status can be measured by a change in . We assume that the change of is abrupt when the process is switched (i.e, a Haines' Jump, Miller and Miller, 1956). Thus, the scale of the change of capillary pressure is modeled by adding a constant change to the macroscopic inverse capillary length scale, . So if we assume that the change of energy status is immediate and abrupt, then the value must jump to another value immediately after the process is switched. Since this model of hysteresis does not change the form of K() and D(), the original analytical solution, given by Eq. [12], still exists. However, the predicted value of from the solution (see Eq. [5]) is scaled by the value of ^-1, which changes depending on the process (drainage, infiltration). This approach is conceptually consistent with the notion of a change in effective pore size associated with the reversal of the flow process (wetting, drying) and the use of as an integrated macroscopic effective capillary length (Philip, 1985; Miller and Miller, 1956).

It is possible to equate our proposed Haines' Jump approach to any model of hysteresis such as the Parlange (1976) model, at least at an integral scale. An analogy is the Green–Ampt integral approximation of the K() and () curves. Studies have indicated K() is non-hysteretic. The parameters K_s and C determine the shape of K(), and K_s is by definition non-hysteretic. Thus, if K() is assumed to be non-hysteretic, the parameter C must be non-hysteretic. According to Eq. [5], assuming no hysteresis of C, the value of the effective inverse macroscopic capillary length scale for drainage (_d) and infiltration (_w) is given by

(13a)

(13b)

where () is a constant non-hysteretic function and _d() and _w() are the drainage and wetting scanning curves, respectively. Solving Eq. [13a] and [13b] for (), and substituting into Eq. [13b] gives

(13c)

It follows that an average effective for any drainage scanning curve, ^*_d, in terms of an effective for infiltration, _w, can be given by

(14)

where _f is the reduced water content at the reversal point. Thus, the value of ^*_d can be predicted a priori for each initial condition, if _w is known (or vice versa). The Haines' Jump in energy status is given by x . Equation [14] can be used with any other existing hysteresis model to estimate a priori the effective Haines' Jump. Two examples, which are subsequently discussed, are models by Parlange (1976) and Mualem (1984). Some hysteresis models may not be compatible with this approach. For example, the Scott et al. (1983) model results in a Haines' Jump that is not reversible (Kool and Parker, 1987). That is, an instantaneous switch from drying to wetting and then back to drying would not leave the value of _d the same. Direct application of hysteresis models such as Parlange (1976) and Mualem (1984) to the Broadbridge and White (1988) hydraulic functions resulted in a modified form of the hydraulic function that invalidates the unified solution of infiltration and drainage (Eq. [12]). However, the unified solution remains valid if a Haines' Jump model is applied to the Broadbridge and White (1988) hydraulic functions. Calculating an effective Haines' Jump using Eq. [14] allows the Parlange (1976) and Mualem (1984) hysteresis models to be applied to the Broadbridge and White (1988) hydraulic functions in an integral sense, which still retains the validity of the unified solution for infiltration and drainage (Eq. [12]). Like Eq. [12], the proposed Haines' Jump hysteresis model can be applied only to situations where the whole profile is draining or the whole profile is wetting.

Parlange Model
The drying and wetting scanning curves can be related by

(15)

where subscripts d and w refer to drying and wetting and subscript i designates the point on the wetting curve where the drying curve is starting. Thus, knowing one scanning curve (_d or _w), the other can be calculated. Comparison with experiments shows that if the shape of the drying scanning curves varies smoothly, then the drying boundary of the loop is sufficient to predict all scanning curves (Parlange, 1976).

The Parlange (1976) hysteresis model has no additional parameter and gives an a priori prediction of the () drying curve from () wetting curve. Unfortunately, the () wetting curve now has a form that does not lend itself to an analytical solution of Richards' equation. However, the functional relationship between _w() and _d() can be substituted into Eq. [14] to calculate numerically an effective Haines' Jump.

Mualem Universal Model
Assuming the distribution functions of water in the pore domains for drying and wetting are the same for the independent domain model, Mualem (1984) presented a universal relationship between the two main curves:

(16)

where _i is the value of pressure head for the starting point of drainage. The predicted drying curve is the lower boundary of the hysteresis domain. For field soil, there may be less pore water blockage against air entry, since there is usually a well-developed structure and a wide range of pore size. In a manner similar to the Parlange (1976) model, it is possible to predict a priori an effective Haines' Jump using Eq. [14].

First-Order Error Analysis
The need to account for hysteresis in hydraulic parameter estimations can be checked by comparing measured water storage during drainage with the water storage during drainage that was predicted using parameters (with uncertainty) obtained from infiltration experiments. Including the effect of uncertainty in the parameters on the estimated soil water storage allows a confidence interval to be placed on predicted water storage. If measured water storage during drainage is outside of the 95% confidence interval of predictions, then the discrepancy cannot be related to uncertainty in parameters estimated from infiltration. Thus, the discrepancy is likely from a change in parameters due to hysteresis.

The mean E[•] and variance var[•] of a function (u) can be derived from its uncertainty parameter vector, u, through a first-order Taylor expansion:

(17)

where is the vector of estimated parameter values, n is the number of parameters, and (u) is the analytical solution of soil water storage (Eq. [12]). Using the expected value operator, E[•], on both side of this expression, we obtain

(18)

Similarly, taking the expectation of ², the variance of - , Var, can be obtained as

(19)

where Cov(u_i,u_j) is the covariance between u_i and u_j. This covariance matrix is usually given by most curve-fitting and inverse procedures. For the linear dependence of (u) upon u, Eq. [18] and [19] are exact. For the nonlinear relationship, Eq. [18] and [19] are good approximations, provided the coefficients of variation of u are small. This first-order analysis provides a way to evaluate the effect of uncertainty in the parameters on the function (u). The derivatives in Eq. [19] were calculated numerically using the software package Mathcad Version 6 (MathSoft, 1995). The estimated parameter vector and the Cov(u_i,u_j) of hydraulic parameters were estimated from measured hydraulic conductivity, K, and pressure head, , as a function of water content during a series of infiltration experiments (Si et al., 1999). The values of and Cov(u_i,u_j) from infiltration were then used with the unified drainage solution (Eq. [12] and [19]) to give confidence intervals for soil water storage (due to parameter uncertainty) during drainage.

	Materials and methods

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Conclusion REFERENCES

 
Field Experiments
Field infiltration experiments were conducted at the Canadian Forces Base Borden, Ontario, Canada, and have been described in detail by Si et al. (1999). Extensive hydrogeological research, including a large-scale, natural-gradient tracer test and a forced-gradient test, has been conducted by the University of Waterloo on this site (Sudicky, 1986). 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 multipurpose TDR probes (Baumgartner et al., 1994; Si et al., 1999). Six 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 metallic cable testers (Tektronix, Wilsonville, OR) by four operators. The readings were taken just before starting the water application and every 5 to 30 min depending on the infiltration rate and the rate of change in , for all 200 multipurpose TDR probes. Here, we use the site average of the 50 probes at a 0.2-m depth as an illustration. At the end of each infiltration rate (i.e., after steady-state water content was established in the soil profile), the water application was stopped and the soil allowed to drain. Soil water content measurements were taken every hour for the initial rapid-drainage phase (10 h) and then taken approximately every 10 h until the soil had drained for 100 h.

A single set of hydraulic parameters with their uncertainty were obtained independently from the infiltration measurements (Si et al., 1999). Since hysteresis in the K() function is assumed to be negligible, the values of K_s and C from the infiltration data were used as known values to estimate a new value of for each drainage event (i.e., _d). The value of _d was obtained using the unified solution (Eq. [12]), inverse procedures, and measurements of soil water storage during drainage. The need to incorporate hysteresis was examined by comparing predicted and measured soil water storage during the drainage and from the change in estimated using infiltration vs. drainage data. The comparison involves an envelope of the uncertainty of water storage introduced by the uncertainty associated with the input parameters and general TDR measurement error (see discussion of Eq. [18]). Finally, measured values of during drainage (i.e., _d) from different initial conditions were compared with effective Haines' Jump values of ^*_d estimated a priori using Eq. [14] and the hysteresis models of Parlange (1976) and Mualem (1984) as examples.

	Results and discussion

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Conclusion REFERENCES

 
Table 1 gives the hydraulic parameters and their correlation matrix estimated from measured K() and () during the infiltration phase of the experiment (Si et al., 1999). Figure 1 shows the measured water storage (0–0.2 m) during drainage for three initial conditions ( , 0.31, and 0.27) and the water storage predicted by directly substituting the hydraulic parameters for infiltration (Table 1) into Eq. [12]. For the wetter initial conditions ( ), the prediction using infiltration parameters underestimates measured water storage during drainage. At the driest initial condition, the measured and predicted values are similar. At , the measured values exceed the upper 95% prediction limits based on parameter uncertainty (from the first-order perturbation approximation). This suggests the differences are from a change in hydraulic parameters (most likely hysteresis). The root mean square error (RMS) of prediction for depth-averaged soil water content was 0.031, 0.014, and 0.015 for , 0.31, and 0.27, respectively. This error is greater than or equal to the average estimated TDR error for absolute soil water content (0.013, Topp et al., 1980). Relative measurement error using TDR, as would be relevant here, would be significantly lower. This also suggests that TDR measurement error cannot account for the discrepancy. In combination, the parameter uncertainty error and TDR measurement error may account for the predicted vs. measured discrepancy.

View larger version (27K): [in this window] [in a new window]   Fig. 1 A comparison of measured soil water storage, predicted soil water storage, and the predicted confidence (95%) interval during drainage using a unified solution with no hysteresis and the value of from infiltration

Figure 2 shows the predicted water storage curves using the best-fit _d value for each of the three initial conditions. The agreement with measured water storage is quite good for all times. The calculated RMSs for depth-averaged water content were 0.0087, 0.006, and 0.006 for , 0.31, and 0.27, respectively; these values are substantially less than the expected measurement error (0.013 m³ m^-3) of TDR (Topp et al., 1980) and much lower than the RMS using _w for infiltration.

View larger version (20K): [in this window] [in a new window]   Fig. 2 A comparison of measured soil water storage and predicted soil water storage during drainage using the Haines' Jump hysteresis model and the value of d from inverse procedures

View larger version (29K): [in this window] [in a new window]   Fig. 3 Comparison of the predicted drying scanning curves from the Parlange (1976) model (dashed line) and corresponding equivalent Haines' Jump hysteresis model (dotted line) for initial water content = 0.38, 0.31, and 0.27. The Broadbridge and White (1988) Eq. [5] as the wetting curve is also shown (solid line)

View larger version (27K): [in this window] [in a new window]   Fig. 4 A comparison of measured soil water storage, predicted soil water storage, and the predicted confidence interval (95%) during drainage using the Haines' Jump hysteresis model and an a priori value of estimated from Eq. [16] and the Parlange (1976) model

For practical purposes, the combination of the proposed Haines' Jump model with the prediction of ^*_d from _w (or vice versa) using Eq. [14] and either the Parlange (1976) model or Mualem (1984) model is likely satisfactory. The RMS error would be about 1 to 2% of water storage, which is within the measurement error of water storage by TDR. This substantiates the conclusion obtained by Jaynes (1985), that simple and complicated hysteresis models usually give similar results. Ignoring hysteresis, however, is unacceptable.

	Conclusion

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Conclusion REFERENCES

 
A Haines' Jump model of hysteresis is proposed and used to create a unified analytical solution for soil water storage to a fixed depth as a function of time during infiltration and drainage. The model accounts for hysteresis by making the inverse macroscopic capillary length scale, , hysteretic. Neglecting hysteresis resulted in poor predictions of water storage during drainage based on hydraulic parameters estimated from infiltration. This was especially true for drainage with high initial water content. Incorporating the proposed hysteresis model resulted in prediction errors less than measurement errors.

A method of a priori estimating the hysteretic nature of was proposed. The method was tested using hysteretic models proposed by Parlange (1976) and Mualem (1984). The predicted hysteresis in was similar to that obtained from best-fit inverse procedures applied independently to soil water storage measurement during infiltration and drainage.

Received for publication February 15, 1999.

	REFERENCES

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Conclusion REFERENCES

 

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• Jaynes, D.B. 1992. Estimation hysteresis in the soil water retention function. In M.Th. van Genuchten, F.J. Leij, and L.J. Lund (ed.) Indirect methods for estimating the hydraulic properties of unsaturated soils. Univ. California, Riverside, CA.
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